Volume 15, Issue 2

We study convergence properties of a new nonlinear Lagrangian method for nonconvex semidefinite programming. The convergence analysis shows that this method converges locally when the penalty parameter is less than a threshold and the error bound of solution is proportional to the penalty parameter under the constraint nondegeneracy condition, the strict complementarity condition and the strong second order sufficient conditions. The major tools used in the analysis include the second implicit function theorem and differentials of Löwner operators.

This paper concerns the study of the numerical approximation for the following initial-boundary value problem: where ƒ : [0, ∞) → [0, ∞) is a C 2 convex, nondecreasing function, and ε is a positive parameter. Under some assumptions, we prove that the solution of a semidiscrete form of the above problem blows up in a finite time and estimate its semidiscrete blow-up time. We also show that the semidiscrete blow-up time in certain cases converges to the real one when the mesh size tends to zero. Finally, we give some numerical experiments to illustrate our analysis.

Recent investigations of M. Rösler [Compos. Math. 143: 749–779, 2007] and M. Voit [J. Theoret. Probab. 22: 741–771, 2009] provide examples of hypergroups with properties similar to the group- or vector space case and with a sufficiently rich structure of automorphisms, providing thus tools to investigate the limit theory of normalized random walks and the structure of the corresponding limit laws. The investigations are parallel to corresponding investigations for vector spaces and simply connected nilpotent Lie groups.

We construct explicit solutions of a system of two conservation laws with small viscosity in the quarter plane {( x, t ) : x > 0, t > 0}, with initial conditions at t = 0 and flux conditions at x = 0. We derive a formula for the limit as viscosity goes to zero which generally belongs to the space of locally bounded Borel measures. This limit satisfies the inviscid equation, in the sense of LeFloch [An existence and uniqueness result for two non-strictly hyperbolic systems, Springer, 1990]. We also treat more general initial and boundary datas and obtain solution in the algebra of generalized functions of Colombeau [C. R. Acad. Sci. Paris Ser. I Math. 317: 851–855, 1993, C. R. Acad. Sci. Paris Ser. I Math. 319: 1179–1183, 1994].

In this paper, we obtain some applications of first order differential subordination and superordination results involving Dziok-Srivastava operator and other linear operators for certain normalized analytic functions in the open unit disk. The results, which are presented in this paper, have relevant connections with various previous results.

In this paper we obtain oscillation criteria for solutions of homogeneous and nonhomogeneous cases of second order neutral differential equations with positive and negative coefficients. Our results improve and extend the results of Manojlović, Shoukaku, Tanigawa and Yoshida [Appl. Math. Comput. 181: 853–863, 2006].