It has been conjectured that for knots K and K′ in S3, w(K#K′) = w(K)+w(K′)−2. In [7], Scharlemann and Thompson proposed potential counterexamples to this conjecture. For every n, they proposed a family of knots {Kn i } for which they conjectured that w(Bn#Kn i ) = w(Kn i ) where Bn is a bridge number n knot. We show that for n > 2 none of the knots in {Kn i } produces such counterexamples.

In this paper we propose a numerical scheme to solve the Kac model of the Boltzmann equation for multiscale rarefied gas dynamics. This scheme is uniformly stable with respect to the Knudsen number, consistent with the fluid-diffusion limit for small Knudsen numbers, and with the Kac equation in the kinetic regime. Our approach is based on the micro-macro decomposition which leads to an equival...

The spectrum of the coherent Rayleigh-Brillouin scattering (CRBS) of light by a rarefied gas is obtained by solving the Boltzmann equation numerically using the fast spectral method. The influence of the intermolecular potential on the CRBS spectrum is investigated and the accuracy of the prevailing Tenti’s s6 kinetic model is evaluated. Our numerical results show that i) the intermolecular pot...

Thermophoresis of axially symmetric bodies is investigated to first order in the Knudsen-mimber, Kn. The study is made in the limit where the typical length of the immersed body is small compared to the mean free path. It is shown that in this case, in contrast to what is the case for spherical bodies, the arising thermal force on the body is not in general antiparallel to the temperature gradi...

Rarefication effects on jet impingement loads are studied by comparing recent new formulas at the collisionless flow limit and numerical simulations. The jet exit size is finite, and can be either planar or round. In the simulations, the jets have different degrees of rarefication, with a Knudsen (Kn) number ranging from 0 to infinity; i.e., the jet flows can be continuum, collisional, or colli...

We have established a simple method for drastically improving the productivity of chemical vapor deposition in large-area graphene synthesis using a roll-stacked Ni coil as a catalyst. Our systematic investigation of the effects of a confined catalytic geometry has shown that the gas flow through interfacial gaps within the stack follows non-continuum fluid dynamics when the size of the gap dec...

In this paper we propose a limit characterization of the behaviour of classes of graphs with respect to their number of spanning trees. Let {Gn} be a sequence of graphs G0,G1,G2, . . . that belong to a particular class. We consider graphs of the form Kn −Gn that result from the complete graph Kn after removing a set of edges that span Gn. We study the spanning tree behaviour of the sequence {Kn...