17 problems

The common-ground conjecture. The pair $\{E,F\}$ is transformable into strong standard canonical form, and the Jordan normal form of its constant nilpotent matrix $N$ consists exac…

Let $E,F:\mathcal I\rightarrow\mathbb{R}^{m\times m}$ be sufficiently smooth, and let $\{E,F\}$ be a regular pair with index $\mu\geq 2$. Write $r$ for the dimension of the differe…

A maximally monotone structure is a structure with the maximal-monotonicity property in the port-Hamiltonian framework. A Dirac structure represents a power-conserving interconnect…

Let $[E,A,B]$ be a singular differential-algebraic system and let $X$ and the constraint-room relation be as defined in the paper. Constraint-room conjecture. The relation … holds…

Let $F(\frac{d}{dz})$ and $E(\frac{d}{dz})$ define a differential operator Dirac structure as above. A differential operator Dirac structure defined by a pair…

Higher-degree preference conjecture. For smooth problems, using a higher degree $N$ is preferable to using a larger number $n$ of subintervals, or equivalently a smaller stepsize…

Let $al V_L$ be the set of load buses, with $P_{si}^0$ and $lpha_i$ denoting the active-power load parameters and $Q_{si}^0$ and $eta_i$ denoting the reactive-power load parameters…

Suppose Assumption 1 and Assumption 2 hold. Consider a differential-algebraic equation system with all loads purely static. Let $al V_t$ denote the generator buses and $al V_L$ the…

Let a power-system differential-algebraic equation model satisfy the assumptions used in the paper, and let the condition obtained for the specific four-bus system be given. Hisken…

Nonlinear splicing conjecture. A similar analysis of splicing conditions is possible for the nonlinear case as well.

Let $N\geq 1$ be the polynomial degree, and let $M$ be the number of collocation points per subinterval in an overdetermined least-squares collocation method for a differential-alg…

Differential-algebraic equations (DAEs) are equations involving differential-algebraic operators, and derivative-array methods are existing numerical methods for their treatment. A…

The paper studies optimal control for differential-algebraic equations, including feedback representations for regular DAEs. Non-regular DAE conjecture. Similar results can also be…

Degree-of-freedom conjecture. A DAE formulation friendly to structural analysis should have a reasonable but never overestimated value $\operatorname{Val}(\boldsymbol{\Sigma})$ tha…

Signature-decrease conjecture. Based on experience, such a decrease tends to give a better formulation of a DAE from the perspective of structural analysis.

Let a differential-algebraic equation system be reformulated by one of the conversion methods described in the paper, and suppose the conversion decreases the value of its signatur…

Consider the system $\Sigma'$ over $\mathrm{T}\mathbb{R}^3$ defined by … It is ruled, meaning that each fiber is a union of lines, so the known necessary condition for flatness doe…