Scheinerman and Wilf [SW94] assert that “an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph Kn.” A rectilinear drawing of Kn is an arrangement of n vertices in the plane, every pair of which is connected by an edge that is a line segment. We assume that no three vertices are collinear, and that no three edges intersec...

We consider the Boltzmann equation for a gas in a horizontal slab, subject to a gravitational force. The boundary conditions are of diffusive type, specifying the wall temperatures, so that the top temperature is lower than the bottom one (Benard setup). We consider a 2-dimensional convective stationary solution, which is close for small Knudsen number to the convective stationary solution of t...

The entropy generation due to steady laminar forced convection fluid flow through parallel plates microchannel is investigated numerically. The effect of Knudsen, Reynolds, Prandtl, Eckert numbers and the nondimensional temperature difference on entropy generation within the microchannel is discussed. The fraction of the entropy generation due to heat transfer to the total entropy generation wi...

The Harary-Hill conjecture states that the minimum number of crossings in a drawing of the complete graph Kn is Z(n) := 1 4 ⌊ n 2 ⌋ ⌊ n−1 2 ⌋ ⌊ n−2 2 ⌋ ⌊ n−3 2 ⌋ . This conjecture was recently proved for 2-page book drawings of Kn. As an extension of this technique, we prove the conjecture for monotone drawings of Kn, that is, drawings where all vertices have different x-coordinates and the edg...

– A non-perturbative analysis of the Bhatnagar-Gross-Krook (BGK) model kinetic equation for finite values of the Knudsen number is presented. This analysis indicates why discrete kinetic versions of the BGK equation, and notably the Lattice Boltzmann method, can provide semi-quantitative results also in the non-hydrodynamic, finite-Knudsen regime, up to Kn ∼ O(1). This may help the interpretati...

We analyze a large number of high-order discrete velocity models for solving the Boltzmann-Bhatnagar-Gross-Krook equation for finite Knudsen number flows. Using the Chapman-Enskog formalism, we prove for isothermal flows a relation identifying the resolved flow regimes for low Mach numbers. Although high-order lattice Boltzmann models recover flow regimes beyond the Navier-Stokes level, we obse...