Some collapsibility results for n-dimensional contingency tables

For a multidimensional contingency table, we obtain several necessary and sufficient conditions for collapsibility and strict collapsibility, using the technique of Möbius inversion formula. As a consequence, the results of Whittemore (Journal of the Royal Statistical Society B, 40, 328–340, 1978) are stated in a form which is easy to understand and the proofs are much simpler and straightforward. Several new results on collapsibility and strict collapsibility with respect to more than one interaction parameter, are established, and their relationships to conditional independence are also pointed out. As applications of our results, several typical examples on collapsibility, strict collapsibility and conditional independence are discussed. It is also shown that Bishop et al. (Discrete Multivariate Analysis: Theory and Practice, MIT Press, Cambridge, 1975) conditions are necessary and sufficient for strict collapsibility with respect to a set of interaction factors.     

Fixed point methods for pseudomonotone variational inequalities involving strict pseudocontractions

We introduce a new iteration method for finding a common element of the set of solutions of a variational inequality problem and the set of fixed points of strict pseudocontractions in a real Hilbert space. The weak convergence of the iterative sequences generated by the method is obtained thanks to improve and extend some recent results under the assumptions that the cost mapping associated with the variational inequality problem only is pseudomonotone and not necessarily inverse strongly monotone. Finally, we present some numerical examples to illustrate the behaviour of the proposed algorithm.     

Operator Equations and Duality Mappings in Sobolev Spaces with Variable Exponents

After studying in a previous work the smoothness of the space UΓ0={u∈W1,p(·)(Ω);u=0 on Γ0 Γ=Ω},where dΓ-measΓ0>0,with p(·)∈C(Ω)and p(x)>1 for all x∈Ω,the authors study in this paper the strict and uniform convexity as well as some special properties of duality mappings defined on the same space.The results obtained in this direction are used for proving existence results for operator equations having the form Ju=Nfu,where J is a duality mapping on UΓ0 corresponding to the gauge function,and Nf is the Nemytskij operator generated by a Carath′eodory function f satisfying an appropriate growth condition ensuring that Nf may be viewed as acting from UΓ0 into its dual.     

Higher Order Efficiency,Saddle Point Optimality,and Duality for Vector Optimization Problems

In this paper, we intend to characterize the strict local efficient solution of order m for a vector minimization problem in terms of the vector saddle point. A new notion of strict local saddle point of higher order of the vector-valued Lagrangian function is introduced. The relationship between strict local saddle point and strict local efficient solution is derived. Lagrange duality is formulated, and duality results are presented.     

Optimization with generalized invexity

Constrained optimization is studied, with nonsmooth (Lipschitz) functions in abstract spaces and cone-constraints. Some more general Lagrangian necessary conditions are obtained, using strict minimum and approximation methods. These conditions are sufficient for a minimum under generalized invex assumptions. A characterization is obtained for generalized invexity, generalizing a known result for differentiable functions. Generalized invexity happens exactly when the generalized Wolfe and Lagrangian dual problems coincide.     

Strict Mittag‐Leffler modules

We characterize strict Mittag‐Leffler modules in terms of free realizations of positive primitive formulas, and rings over which (pure‐) projectives are trivial in terms of various notions of separability of (flat) strict Mittag‐Leffler modules. © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

NCP Functions Applied to Lagrangian Globalization for the Nonlinear Complementarity Problem

Based on NCP functions, we present a Lagrangian globalization (LG) algorithm model for solving the nonlinear complementarity problem. In particular, this algorithm model does not depend on some specific NCP function. Under several theoretical assumptions on NCP functions we prove that the algorithm model is well-defined and globally convergent. Several NCP functions applicable to the LG-method are analyzed in details and shown to satisfy these assumptions. Furthermore, we identify not only the properties of NCP functions which enable them to be used in the LG method but also their properties which enable the strict complementarity condition to be removed from the convergence conditions of the LG method. Moreover, we construct a new NCP function which possesses some favourable properties.     

Strictly positive definite functions on the unit circle

We study strictly positive definite functions on the unit circle in the Euclidean space of dimension two. We develop several conditions pertaining to the determination of such functions. The major result is obtained by considering the set of real numbers as a vector space over the field of rational numbers and then applying the Kronecker approximation theorem and Weyl's criterion on equidistributions.