We provide a new lower bound on the number of (≤ k)-edges of a set of n points in the plane in general position. We show that for 0 ≤ k ≤ bn−2 2 c the number of (≤ k)-edges is at least Ek(S) ≥ 3 ( k + 2 2 ) + k ∑ j=b3 c (3j − n + 3), which, for b3 c ≤ k ≤ 0.4864n, improves the previous best lower bound in [11]. As a main consequence, we obtain a new lower bound on the rectilinear crossing numbe...

For double arrays of constants {ani, 1 ≤ i ≤ kn, n ≥ 1} and sequences of negatively orthant dependent random variables {Xn, n ≥ 1}, the conditions for strong law of large number of ∑kn i=1 aniXi are given. Both cases kn ↑ ∞ and kn = ∞ are treated.

For some given graph H, a G is called H-covered if each vertex in contained copy of H. In this note, we determine the maximum number independent sets size t ≥ 3 N-vertex K n -covered graphs and classify extremal graphs. The result answers question proposed by Chakraborti Loh. proof uses an edge-switching operation on hypergraphs which never increases sets.

Knudsen and Preneel (Asiacrypt’96 and Crypto’97) introduced a hash function design in which a linear error-correcting code is used to build a wide-pipe compression function from underlying blockciphers operating in Davies-Meyer mode. In this paper, we (re)analyse the preimage resistance of the Knudsen-Preneel compression functions in the setting of public random functions. We give a new non-ada...

We introduce a scheme which gives rise to additional degree of freedom for the same number of discrete velocities in the context of the lattice Boltzmann model. We show that an off-lattice D3Q27 model exists with correct equilibrium to recover Galilean-invariant form of Navier-Stokes equation (without any cubic error). In the first part of this work, we show that the present model can capture t...

Direct simulation Monte Carlo (DSMC) of shock interaction in hypersonic low density flow is developed. Three collision molecular models, including hard sphere (HS), variable hard sphere (VHS), and variable soft sphere (VSS), are employed in the DSMC study. The simulations of double-cone and Edney's type IV hypersonic shock interactions in low density flow are performed. Comparisons between DSMC...

The hydrodynamic limit for the Boltzmann equation is studied in the case when the limit system, that is, the system of Euler equations contains contact discontinuities. When suitable initial data is chosen to avoid the initial layer, we prove that there exists a unique solution to the Boltzmann equation globally in time for any given Knudsen number. And this family of solutions converge to the ...

It is experimentally known that a pressure di¤erence is developed in a gas under continuum conditions contained in a closed capillary tube, the ends of which are maintained at di¤erent temperatures. However, in contradiction with this experimentally measurable thermomolecular pressure di¤erence, the use of the methods of irreversible thermodynamics in conjunction with the Navier-Stokes equation...

For n ≤ 27 we present exact values for the maximum number h(n) of halving lines and eh(n) of halving pseudolines, determined by n points in the plane. For this range of values of n we also present exact values of the rectilinear cr(n) and the pseudolinear e cr(n) crossing number of the complete graph Kn. eh(n) and e cr(n) are new for n ∈ {14, 16, 18, 20, 22, 23, 24, 25, 26, 27}, h(n) is new for...