That is, in both cases we construct n−1 pairwise-commuting partial differential operators which, in turn, commute with the respective quantum Hamiltonians.
Moreover, the joint eigenvalues of the commuting partial differential operators appear to be encoded linearly as parameters in an n-parameter eigenvalue equation given by a second-order complex ordinary differential equation with automorphic boundary conditions.
Then (Delta ) commutes with the operators, (partial ), (overline{partial }), (partial ^*), (overline{partial }^*), L, (Lambda ).
Then (Delta ) commutes with the operators, (partial ), (overline{partial }), (partial ^*), (overline{partial }^*), L, (Lambda ). (ii) In particular, a k- form (eta = sum _{p+q = k} eta _{p,q}) is harmonic if and only if all the (eta _{p,q})'s are harmonic.
We present an alternating-direction implicit (ADI) finite difference formulation for space-fractional diffusion equations in three space dimensions and prove its unconditional stability and convergence rate provided that the fractional partial difference operators along x-, y-, z-directions commute.
One of the five state Supreme Court justices who declined in January to block the execution issued a partial dissent calling for Governor Sundquist, a Republican, to commute the sentence because of "extenuating circumstances".
The vectorial continued fraction (1.1) where its terms are commuting elements in diverges, if its odd partial denominators are all quasinilpotent elements in.
This article attempts to fill this gap by examining four commuting benchmarks (minimum commute, maximum commute, random commute, and proportionally matched commute) and five excess commuting indices (excess commute, commuting potential utilized, commuting economy, normalized commuting economy, and effort).
