Asymptotic approximation for the solution to a semilinear parabolic problem in a thin star‐shaped junction

A semilinear parabolic problem is considered in a thin 3‐D star‐shaped junction that consists of several thin curvilinear cylinders that are joined through a domain (node) of diameter The purpose is to study the asymptotic behavior of the solution u_ε as ε→0, ie, when the star‐shaped junction is transformed in a graph. In addition, the passage to the limit is accompanied by special intensity factors and in nonlinear perturbed Robin boundary conditions. We establish qualitatively different cases in the asymptotic behavior of the solution depending on the value of the parameters {α_i}and {β_i}. Using the multiscale analysis, the asymptotic approximation for the solution is constructed and justified as the parameter ε→0. Namely, in each case, we derive the limit problem (ε=0)on the graph with the corresponding Kirchhoff transmission conditions (untypical in some cases) at the vertex, define other terms of the asymptotic approximation and prove appropriate asymptotic estimates that justify these coupling conditions at the vertex, and show the impact of the local geometric heterogeneity of the node and physical processes in the node on some properties of the solution.     

Spectral analysis for the Gauss problem on continued fractions

We present a new derivation of the formula appearing in Babenko (1978) and Mayer and Roepstorff (1987) that gives the probability distribution of τ⁻ⁿ${\tau }^{-n}$ in terms of the eigenvalues of a symmetric operator. Here τ$\tau$ is the well-known Gauss-map.     

An inexact bundle variant suited to column generation

We give a bundle method for constrained convex optimization. Instead of using penalty functions, it shifts iterates towards feasibility, by way of a Slater point, assumed to be known. Besides, the method accepts an oracle delivering function and subgradient values with unknown accuracy. Our approach is motivated by a number of applications in column generation, in which constraints are positively homogeneous—so that zero is a natural Slater point—and an exact oracle may be time consuming. Finally, our convergence analysis employs arguments which have been little used so far in the bundle community. The method is illustrated on a number of cutting-stock problems. Research supported by INRIA New Investigation Grant “Convex Optimization and Dantzig–Wolfe Decomposition”.     

Valid inequalities for MIPs and group polyhedra from approximate liftings

In this paper, we present an approximate lifting scheme to derive valid inequalities for general mixed integer programs and for the group problem. This scheme uses superadditive functions as the building block of integer and continuous lifting procedures. It yields a simple derivation of new and known families of cuts that correspond to extreme inequalities for group problems. This new approximate lifting approach is constructive and potentially efficient in computation. J.-P. P. Richard was supported by NSF grant DMI-348611.     

The Riemann–Hilbert problem to coupled nonlinear Schrödinger equation: Long-time dynamics on the half-line

We derive the long-time asymptotics for the solution of initial-boundary value problem of coupled nonlinear Schrödinger equation whose Lax pair involves 3 × 3 matrix in present paper. Based on a nonlinear steepest descent analysis of an associated 3 × 3 matrix Riemann–Hilbert problem, we can give the precise asymptotic formulas for the solution of the coupled nonlinear Schrödinger equation on the half-line.     

Growth Behavior of Two Classes of Merit Functions for Symmetric Cone Complementarity Problems

In the solution methods of the symmetric cone complementarity problem (SCCP), the squared norm of a complementarity function serves naturally as a merit function for the problem itself or the equivalent system of equations reformulation. In this paper, we study the growth behavior of two classes of such merit functions, which are induced by the smooth EP complementarity functions and the smooth implicit Lagrangian complementarity function, respectively. We show that, for the linear symmetric cone complementarity problem (SCLCP), both the EP merit functions and the implicit Lagrangian merit function are coercive if the underlying linear transformation has the P-property; for the general SCCP, the EP merit functions are coercive only if the underlying mapping has the uniform Jordan P-property, whereas the coerciveness of the implicit Lagrangian merit function requires an additional condition for the mapping, for example, the Lipschitz continuity or the assumption as in (45). The authors would like to thank the two anonymous referees for their helpful comments which improved the presentation of this paper greatly. The research of J.-S. Chen was partially supported by National Science Council of Taiwan.     

On the solutions of the equation AXB = C under Toeplitz‐like and Hankel matrices constraint

In this paper, we are mainly concerned with 2 types of constrained matrix equation problems of the form AXB=C, the least squares problem and the optimal approximation problem, and we consider several constraint matrices, such as general Toeplitz matrices, upper triangular Toeplitz matrices, lower triangular Toeplitz matrices, symmetric Toeplitz matrices, and Hankel matrices. In the first problem, owing to the special structure of the constraint matrix , we construct special algorithms; necessary and sufficient conditions are obtained about the existence and uniqueness for the solutions. In the second problem, we use von Neumann alternating projection algorithm to obtain the solutions of problem. Then we give 2 numerical examples to demonstrate the effectiveness of the algorithms.     

A Fokas approach to the coupled modified nonlinear Schrödinger equation on the half‐line

In this work, we discuss the coupled modified nonlinear Schrödinger (CMNLS) equation, which describe the pulse propagation in the picosecond or femtosecond regime of the birefringent optical fibers. By use of the Fokas approach, the initial‐boundary value problem for the CMNLS equation related to a 3×3 matrix Lax pair on the half‐line is to be analyzed. Assuming that the solution {u(x,t),v(x,t)} of CMNLS equation exists, we will prove that it can be expressed in terms of the unique solution of a 3×3 matrix Riemann‐Hilbert problem formulated in the plane of the complex spectral parameter λ. Moreover, we also get that some spectral functions s(λ) and S(λ) are not independent of each other but meet a global relationship.     

Inf‐sup stability of Petrov‐Galerkin immersed finite element methods for one‐dimensional elliptic interface problems

In this article, we analyze the Petrov‐Galerkin immersed finite element method (PG‐IFEM) when applied to one‐dimensional elliptic interface problems. In the PG‐IFEM (T. Hou, X. Wu and Y. Zhang, Commun. Math. Sci., 2 (2004), 185‐205, and S. Hou and X. Liu, J. Comput. Phys., 202 (2005), 411‐445), the classic immersed finite element (IFE) space was taken as the trial space while the conforming linear finite element space was taken as the test space. We first prove the inf‐sup condition of the PG‐IFEM and then show the optimal error estimate in the energy norm. We also show the optimal estimate of the condition number of the stiffness matrix. The results are extended to two dimensional problems in a special case.     

Nonstationary Maxwell equations

ABSTRACT

The paper deals with a mixed problem for nonstationary generalised Maxwell equations. The boundary conditions are of Riemann-Hilbert type. The problem is reduced to a mixed problem for a wave equation where the boundary conditions are of Dirichlet type as they were introduced by D. Spencer in the middle 1950?s. We use the Fourier method to construct an approximate solution to the problem in certain function spaces of Sobolev type.