A new class of Ramsey-Turán problems

We introduce and study a new class of Ramsey-Turán problems, a typical example of which is the following one:Let ε>0 and G be a graph of sufficiently large order n with minimum degree δ(G)>3n/4. If the edges of G are colored in blue or red, then for all k∈[4,⌊(1/8−ε)n⌋], there exists a monochromatic cycle of length k.     

Predictor-corrector algorithm for solvingP *(κ)-matrix LCP from arbitrary positive starting points

A new predictor-corrector algorithm is proposed for solvingP _*(κ)-matrix linear complementarity problems. If the problem is solvable, then the algorithm converges from an arbitrary positive starting point (x ⁰,s ⁰). The computational complexity of the algorithm depends on the quality of the starting point. If the starting point is feasible or close to being feasible, it has -iteration complexity, whereρ ₀ is the ratio of the smallest and average coordinate ofX ⁰ s ⁰. With appropriate initialization, a modified version of the algorithm terminates in O((1+κ)²(n/ρ ₀)L) steps either by finding a solution or by determining that the problem has no solution in a predetermined, arbitrarily large, region. The algorithm is quadratically convergent for problems having a strictly complementary solution. We also propose an extension of a recent algorithm of Mizuno toP _*(κ)-matrix linear complementarity problems such that it can start from arbitrary positive points and has superlinear convergence without a strictly complementary condition. The work of this author was supported in part by NSF, Grant DMS 9305760 and by an Oberman fellowship from the University of Iowa Center for Advanced Studies.     

Analysis of expanded mixed methods for fourth-order elliptic problems

The recently proposed expanded mixed formulation for numerical solution of second-order elliptic problems is here extended to fourth-order elliptic problems. This expanded formulation for the differential problems under consideration differs from the classical formulation in that three variables are treated, i.e., the displacement, the stress, and the moment tensors. It works for the case where the coefficient of the differential equations is small and does not need to be inverted, or for the case in which the stress tensor of the equations does not need to be symmetric. Based on this new formulation, various mixed finite elements for fourth-order problems are considered; error estimates of quasi-optimal or optimal order depending upon the mixed elements are derived. Implementation techniques for solving the linear system arising from these expanded mixed methods are discussed, and numerical results are presented. © 1997 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 13: 483–503, 1997     

Back-and-forth shooting method for solving two-point boundary-value problems

The paper proposes a special iterative method for a nonlinear TPBVP of the form (t)=f(t, x(t),p(t)), (t)=g(t, x(t),p(t)), subject toh(x(0),p(0))=0,e(x(T),p(T))=0. Certain stability properties of the above differential equations are taken into consideration in the method, so that the integration directions associated with these equations respectively are opposite to each other, in contrast with the conventional shooting methods. Via an embedding and a Riccati-type transformation, the TPBVP is reduced to consecutive initial-value problems of ordinary differential equations. A preliminary numerical test is given by a simple example originating in an optimal control problem.     

Boundary integral equation method in micropolar elasticity

This paper presents the integral representations of the displacement and rotation fields of the micropolar continuum, and the regularized integral representations of the traction and couple vector. Thus the boundary integral equations and the boundary integro-differential equations are derived. In addition, the formulation for crack problems is given by both the boundary integral equations and the boundary integro-differential equations.     

Target set reachability criteria for dynamical systems described by inaccurate models

S is taken to be a dynamical system (described by Banach space operators) whose outputy we wish to regulate. The structural complexity ofS (nonlinearities, distributed parameters, etc.) forces us to design a controller forS using an approximate modelM ofS. A convex error bound ? describes the accuracy of the approximation ofS byM. For a prescribed target setY _t , we considered the problem of driving the output ofS toY _t subject to worst possible error excursions betweenM andS. The notion of areconstructed support function is instrumental to the derivation of the main result, Theorem 6.1, which we can paraphrase as follows. IfM is linear (S need not be), then we can describe a finite-dimensional convex programming Problem (P), whose solution tells us whether or notY _t is reachable. Theorem 6.1 is then specialized to differential systems approximated in the norm. The computation of numerical solutions is also discussed.     

An analysis of overlapping Schwarz method for a weakly coupled system of singularly perturbed convection-diffusion equations

In this article, we have developed an overlapping Schwarz method for a weakly coupled system of convection-diffusion equations. The method splits the original domain into two overlapping subdomains. A hybrid difference scheme is proposed in which on the boundary layer region, we use the central finite difference scheme on a uniform mesh, whereas on the nonlayer region, we use the mid-point difference scheme on a uniform mesh. It is shown that the numerical approximations converge in the maximum norm to the exact solution. We have proved that, when appropriate subdomains are used, the method produces almost second-order convergence. Furthermore, it is shown that two iterations are sufficient to achieve the expected accuracy. Numerical examples are presented to support the theoretical results. The main advantage of this method used with the proposed scheme is that it reduces iteration counts very much and easily identifies in which iteration the Schwarz iterate terminates.     

The eigenvalue complementarity problem

In this paper an eigenvalue complementarity problem (EiCP) is studied, which finds its origins in the solution of a contact problem in mechanics. The EiCP is shown to be equivalent to a Nonlinear Complementarity Problem, a Mathematical Programming Problem with Complementarity Constraints and a Global Optimization Problem. A finite Reformulation–Linearization Technique (Rlt)-based tree search algorithm is introduced for processing the EiCP via the lattermost of these formulations. Computational experience is included to highlight the efficacy of the above formulations and corresponding techniques for the solution of the EiCP.     

On expansions of functions of two variables in mixed Fourier-Jacobi series and their application for estimation of errors of cubature formulas

Some problems of expansion of functions of two variables in mixed Fourier-Jacobi series are discussed. In particular, estimates of the convergence rate of these series on classes of functions of two variables characterized by generalized moduli of continuity are given. Applications of these results and estimation of residues of some Chebyshev-type mixed cubature formulas are discussed.     

A Discrete Phenomenon in the C and Analytic Regularity of the Blow-Up Curve of Solutions to the Liouville Equation in One Space Dimension

We give a complete discussion of the C^∞ or analytic regularity of blow-up curves for Cauchy problems or some mixed problems for the Liouville equation in one space dimension. In the case of mixed problems, the regularity results depend on the boundary condition: actually, we show the existence of a sequence of boundary conditions for which the regularity of the blow-up curve is better than in the general case.