INTERPOLATION WITH LAGRANGE POLYNOMIALS A SIMPLE PROOF OF MARKOV INEQUALITY AND SOME OF ITS GENERALIZATIONS

The following theorem is proved Theorem 1.Let q be a polynomial of degree n(qP_n)with n distinct zeroes lying in the interval[-1,1] and △'_q={-1}∪{τ_i:q'(τ_i)=0,i=1,n-1}∪{1}. If polynomial pP_n satisfies the inequality then for each k=1,n and any x[-1,1]its k-th derivative satisfies the inequality 丨p^(k)(x)丨≤max{丨q^((k))(x)丨,丨1/k(x^2-1)q^(k+1)(x)+xq^((k))(x)丨}. This estimate leads to the Markov inequality for the higher order derivatives of polynomials if we set q=T_n,where Tn is Chebyshev polynomial least deviated from zero. Some other results are established which gives evidence to the conjecture that under the conditions of Theorem 1 the inequality ‖p^((k))‖≤‖q^(k)‖holds.

A SIMPLE PROOF OF THE INEQUALITY FFD (L)≤11/9 OPT(L)+1, ■L FOR THE FFD BIN-PACKING ALGORITHM

The first fit decreasing (FFD) heuristic algorithm is one of the most famous and moststudied methods for an approximative solution of the bin-packing problem. For a list L, letOPT(L) denote the minimal number of bins into which L can be packed, and let FFD(L)denote the number of bins used by FFD. Johnson showed that for every list L, FFD(L)≤11/9OPT(L)+4. His proof required more than 100 pages. Later, Baker gave a much shorterand simpler proof for FFD(L)≤11/9OPT(L)+3. His proof required 22 pages. In this paper,we give a proof for FFD(L)≤11/9 OPT(L)+1. The proof is much simpler than the previousones.

Cluster synchronization of community network with distributed time delays via impulsive control

Cluster synchronization is an important dynamical behavior in community networks and deserves further investigations.A community network with distributed time delays is investigated in this paper.For achieving cluster synchronization,an impulsive control scheme is introduced to design proper controllers and an adaptive strategy is adopted to make the impulsive controllers unified for different networks.Through taking advantage of the linear matrix inequality technique and constructing Lyapunov functions,some synchronization criteria with respect to the impulsive gains,instants,and system parameters without adaptive strategy are obtained and generalized to the adaptive case.Finally,numerical examples are presented to demonstrate the effectiveness of the theoretical results.

Solving two-dimensional Volterra-Fredholm integral equations of the second kind by using Bernstein polynomials

In this paper, we present a numerical method for solving two-dimensional VolterraFredholm integral equations of the second kind（2DV-FK2）. Our method is based on approximating unknown function with Bernstein polynomials. We obtain an error bound for this method and employ the method on some numerical tests to show the efficiency of the method.

A SIMPLE PROOF OF THE INEQUALITY R_M(MF(κ))≤1.2+(1/2~κ) IN MULTIPROCESSOR SCHEDULING

We consider the well-known problem of scheduling n independent tasks nonpreemptivelyon m identical processors with the objective of minimizing the makespan. Coffman, Garey andJohnson described an algorithm MULTIFIT, based on bin-packing, with a worst case performancebetter than the LPT-algorithm. The bound 1.22 obtained by them was claimed by Friesen in1984 that it can be improved to 1.2. In this paper we give a simp1e proof for this bound.

Global well-posedness of 3-D density-dependent Navier-Stokes system with variable viscosity

Given initial data(ρ0, u0) satisfying 0 < m ρ0≤ M, ρ0- 1 ∈ L2∩˙W1,r(R3) and u0 ∈˙H-2δ∩ H1(R3) for δ∈ ]1/4, 1/2[ and r ∈ ]6, 3/1- 2δ[, we prove that: there exists a small positive constant ε1,which depends on the norm of the initial data, so that the 3-D incompressible inhomogeneous Navier-Stokes system with variable viscosity has a unique global strong solution(ρ, u) whenever‖ u0‖ L2 ‖▽u0 ‖L2 ≤ε1 and ‖μ(ρ0)- 1‖ L∞≤ε0 for some uniform small constant ε0. Furthermore, with smoother initial data and viscosity coefficient, we can prove the propagation of the regularities for such strong solution.

Endpoint Estimates for Marcinkiewicz Integrals on Weighted Weak Hardy Spaces

In this paper, we obtain the（W Hω^1, W Lω^1） type estimate for the Marcinkiewicz integral and the（W H1 b,ω, W L1ω） type estimate for the commutator generated by a BMO function and the Marcinkiewicz integral, where the kernel satisfies a certain logarithmic type Lipschitz condition.

The Cauchy Problem for Coupled Nonlinear Schrdinger Equations with Linear Damping： Local and Global Existence and Blowup of Solutions

The authors study, by applying and extending the methods developed by Cazenave(2003), Dias and Figueira(2014), Dias et al.(2014), Glassey(1994–1997), Kato(1987), Ohta and Todorova(2009) and Tsutsumi(1984), the Cauchy problem for a damped coupled system of nonlinear Schrdinger equations and they obtain new results on the local and global existence of H^1-strong solutions and on their possible blowup in the supercritical case and in a special situation, in the critical or supercritical cases.

An Integral Representation for the Weighted Geometric Mean and Its Applications

By virtue of Cauchy’s integral formula in the theory of complex functions,the authors establish an integral representation for the weighted geometric mean,apply this newly established integral representation to show that the weighted geometric mean is a complete Bernstein function,and find a new proof of the well-known weighted arithmetic-geometric mean inequality.