Linked partitions and linked cycles

The notion of noncrossing linked partition arose from the study of certain transforms in free probability theory. It is known that the number of noncrossing linked partitions of [n+1] is equal to the n-th large Schröder number r_n, which counts the number of Schröder paths. In this paper we give a bijective proof of this result. Then we introduce the structures of linked partitions and linked cycles. We present various combinatorial properties of noncrossing linked partitions, linked partitions, and linked cycles, and connect them to other combinatorial structures and results, including increasing trees, partial matchings, k-Stirling numbers of the second kind, and the symmetry between crossings and nestings over certain linear graphs.     

The unstable spectrum of the Navier–Stokes operator in the limit of vanishing viscosity

The Navier–Stokes equations for the motion of an incompressible fluid in three dimensions are considered. A partition of the evolution operator into high frequency and low frequency parts is derived. This decomposition is used to prove that the eigenvalues of the Navier–Stokes operator in the inviscid limit converge precisely to the eigenvalues of the Euler operator beyond the essential spectrum.     

Cyclohexanecarboxamide

The title compound, C₇H₁₃NO, forms R²₂(8) N—H...O hydrogen‐bonded dimers and C4 N—H...O‐linked chains, which are further stabilized by a C—H...O interaction. The combination of these interactions results in a hydrogen‐bonded network parallel to (100), with a motif that can be described by the secondary graph set R⁴₆(16). The existence of the same hydrogen‐bonding motif in 1‐phenylcyclopentanecarboxamide and 1‐(2‐bromophenyl)cyclohexanecarboxamide [Lemmerer & Michael (2008). CrystEngComm, 10 , 95–102 indicates that replacing the H atom on position 1 with a more bulky group does not necessarily disrupt the observed hydrogen‐bonding pattern. The presence of a C—H...O interaction to stabilize the R⁴₆(16) network does, however, seem to be required. In addition, the title compound is isomorphous with a previously published structure of cyclopentanecarboxamide [Winter et al. (1981). Acta Cryst. B 37 , 2183–2185].