Issue 2, Volume 43, June 2018

2018

1. Differential Stability of Convex Discrete Optimal Control Problems Duong Thi Viet An , Nguyen Thi Toan PDF

   ▸ Abstract

   Differential stability of convex discrete optimal control problems in Banach spaces is studied in this paper. By using some recent results of An and Yen (Appl. Anal. 94, 108–128, 2015) on differential stability of parametric convex optimization problems under inclusion constraints, we obtain an upper estimate for the subdifferential of the optimal value function of a parametric convex discrete optimal control problem, where the objective function may be nondifferentiable. If the objective function is differentiable, the obtained upper estimate becomes an equality. It is shown that the singular subdifferential of the just mentioned optimal value function always consists of the origin of the dual space.

   201-217
2. Semilinear Hyperbolic Boundary Value Problem Associated to the Nonlinear Generalized Viscoelastic Equations Abita Rahmoune PDF

   ▸ Abstract

   Consider a semilinear hyperbolic boundary value problem associated to the nonlinear generalized viscoelastic equations with Direchlet-Neumann boundary conditions. Then, the global existence of a weak solution is established. The uniqueness of the solution has been obtained by eliminating some hypotheses that have been imposed by other authors for different particular problems.

   219-238
3. A New Simple Parallel Iteration Method for a Class of Variational Inequalities Nguyen Song Ha , Nguyen Buong , Nguyen Thi Thu Thuy PDF

   ▸ Abstract

   In this paper, we propose a new simple parallel iterative method to find a solution for variational inequalities over the set of common fixed points of an infinite family of nonexpansive mappings on real reflexive and strictly convex Banach spaces with a uniformly Gâteaux differentiable norm. Our parallel iterative method is simpler than the one proposed by Buong et al. (Numer. Algorithms 72, 467–481 2016). An iterative method of Halpern type for common zeros of an infinite family of $m$-accretive mappings is shown as a special case of our result. Two numerical examples are also given to illustrate the effectiveness and superiority of the proposed algorithm.

   239-255
4. Unique Path Lifting from Homotopy Point of View Mehdi Tajik , Behrooz Mashayekhy , Ali Pakdaman PDF

   ▸ Abstract

   The paper introduces some notions extending the unique path lifting property from a homotopy viewpoint and studies their roles in the category of fibrations. First, we define some homotopical kinds of the unique path lifting property and find all possible relationships between them. Moreover, we supplement the full relationships of these new notions in the presence of fibrations. Second, we deduce some results in the category of fibrations with these notions instead of unique path lifting such as the existence of products and coproducts. Also, we give a brief comparison of these new categories to some categories of the other generalizations of covering maps. Finally, we present two subgroups of the fundamental group related to the fibrations with these notions and compare them to the subgroups of the fundamental group related to covering and generalized covering maps.

   257-273
5. On the Length Function of Saturations of Ideal Powers Đoàn Trung Cường , Phạm Hồng Nam , Phạm Hùng Quý PDF

   ▸ Abstract

   For an ideal $I$ in a Noetherian local ring $(R, \mathfrak{m})$, we prove that the integer-valued function $\ell_{R}(H^{0}_{\mathfrak{m}}(R/I^{n + 1}))$ is a polynomial for $n$ big enough if either $I$ is a principal ideal or $I$ is generated by part of an almost $p$-standard system of parameters and $R$ is unmixed. Furthermore, we are able to compute the coefficients of this polynomial in terms of length of certain local cohomology modules and usual multiplicity if either the ideal is principal or it is generated by part of a standard system of parameters in a generalized Cohen-Macaulay ring. We also give an example of an ideal generated by part of a system of parameters such that the function $\ell_{R}(H^{0}_{\mathfrak{m}} (R/I^{n + 1}))$ is not a polynomial for $n \gg 0$.

   275-288
6. Piecewise Linear Vector Optimization Problems on Locally Convex Hausdorff Topological Vector Spaces Nguyen Ngoc Luan PDF

   ▸ Abstract

   Piecewise linear vector optimization problems in the locally convex Hausdorff topological vector space setting are considered in this paper. The efficient solution set of these problems are shown to be the unions of finitely many semi-closed generalized polyhedral convex sets. If, in addition, the problem is convex, then the efficient solution set and the weakly efficient solution set are the unions of finitely many generalized polyhedral convex sets and they are connected by line segments. Our results develop the preceding ones of Zheng and Yang (Sci. China Ser. A. 51, 1243–1256 2008), and Yang and Yen (J. Optim. Theory Appl. 147, 113–124 2010), which were established in the normed space setting.

   289-308
7. On Permutations Induced by Tame Automorphisms Over Finite Fields Keisuke Hakuta PDF

   ▸ Abstract

   The present paper deals with permutations induced by tame automorphisms over finite fields. The first main result is a formula for determining the sign of the permutation induced by a given elementary automorphism over a finite field. The second main result is a formula for determining the sign of the permutation induced by a given affine automorphism over a finite field. We also give a combining method of the above two formulae to determine the sign of the permutation induced by a given triangular automorphism over a finite field. As a result, for a given tame automorphism over a finite field, if we know a decomposition of the tame automorphism into a finite number of affine automorphisms and elementary automorphisms, then one can easily determine the sign of the permutation induced by the tame automorphism.

   309-324
8. Some Stability Properties of Parametric Quadratically Constrained Nonconvex Quadratic Programs in Hilbert Spaces Vu Van Dong PDF

   ▸ Abstract

   Stability of nonconvex quadratic programming problems under finitely many convex quadratic constraints in Hilbert spaces is investigated. We present several stability properties of the global solution map, and the continuity of the optimal value function, assuming that the problem data undergo small perturbations.

   325-340
9. On a Backward Heat Conduction Problem Associated with Asymmetric Final Data Le Minh Triet , Luu Hong Phong , Pham Hoang Quan PDF

   ▸ Abstract

   In this work, we investigate an abstract parabolic problem associated with the final data, with one specific case of the abstract parabolic problem being considered in polar coordinates. Specially, the present paper gives the first treatment which is not only more general but also more applicable in the case the backward heat problem (BHP) is not symmetric and not axisymmetric in polar coordinates. In general, the above problem is severely ill-posed in the sense of Hadamard. Here, we propose the modified quasi-boundary value (MQBV) method in order to obtain the stability of the regularized solution for the problem. To our knowledge, our results are new and they improve the results of two previous papers (see Cheng and Fu (Inverse Probl. Sci. Eng. 17(8), 1085–1093 2009), (Acta Math. Sinica, English Series 26(11), 2157–2164 2010)) while the authors only considered axisymmetric or radially symmetric data. Finally, a numerical example is given to demonstrate the feasibility and efficiency of our method.

   341-356
10. Codes Induced by Alternative Codes Ngo Thi Hien , Do Long Van PDF

    ▸ Abstract

    Alternative codes, an extension of the notion of ordinary codes, have been first introduced and considered by Huy and Nam (2004). As seen below, every alternative code, in its turn, defines an ordinary code. Such codes are called codes induced by alternative codes or alt-induced codes, for short. In this paper, we consider these alt-induced codes and subclasses of them. In particular, characteristic properties of such codes are established, and an algorithm to check whether a finite code is alt-induced or not is proposed.

    357-371
11. Quasi-interpolation Representation and Sampling Recovery of Multivariate Functions Nguyen Manh Cuong , Mai Xuan Thao PDF

    ▸ Abstract

    In this paper, we prove a multivariate generalization of the quasi-interpolation representation of functions in Besov-type spaces by B-spline series with some equivalent discrete quasi-norms. Moreover, by using this representation, we construct linear sampling methods which give the asymptotic order of optimal linear sampling methods.

    373-389