Issue 1, Volume 42, March 2017

2017

1. A New Method for Determining all Maximal Efficient Faces in Multiple Objective Linear Programming Ta Van Tu PDF

   ▸ Abstract

   Most of the known methods for finding the efficient set of a multiple objective linear programming (MOLP) problem are bottom-up search methods. Main difficulties of the known bottom-up search methods are to find all efficient extreme points adjacent to and to enumerate all efficient faces incident to an efficient degenerate extreme point. Main drawbacks of these methods are that the computational cost is still large and an implementation of them is still difficult. In this paper we propose a new local bottom-up search method for finding all maximal efficient faces for an MOLP problem. Our method is based on the maximal descriptor index sets for efficient edges and extreme rays for the MOLP problem in which the maximal descriptor index sets for edges and extreme rays incident to an efficient degenerate extreme point are easily found on the basis of solving some special linear programming problems. In addition, all efficient extreme points adjacent to and all efficient faces incident to an efficient extreme point can be easily found without using the simplex tables corresponding to bases of this point. Our method can overcome difficulties caused by the degeneracy of faces and is easy to implement. Some comparisons of our method with the known bottom-up search methods are presented. A numerical example is given to illustrate the method.

   1-25
2. On $\phi$-2-Absorbing Primary Submodules Razieh Moradi , Mahdieh Ebrahimpour PDF

   ▸ Abstract

   Let $R$ be a commutative ring with identity and $M$ be a unitary $R$-module. Let $\phi :S(M)\rightarrow S(M)\cup \{\emptyset \}$ be a function, where $S(M)$ is the set of submodules of $M$. We say that a proper submodule $N$ of $M$ is a $\phi$-2-absorbing primary submodule if $rsx\in N∖\phi(N)$ implies $r x\in N$, or $s x\in N$, or $rs\in \sqrt {(N:M)}$, where $r,s\in R$ and $x\in M$. In this paper, we study $\phi$-2-absorbing primary submodules and we prove some basic properties of these submodules. Also, we give a characterization of $\phi$-2-absorbing primary submodules and we investigate $\phi$-2-absorbing primary submodules of some well-known modules.

   27-35
3. Local Cohomology Annihilators and Macaulayfication Nguyen Tu Cuong , Doan Trung Cuong PDF

   ▸ Abstract

   The aim of this paper is to study a deep connection between local cohomology annihilators and Macaulayfication and arithmetic Macaulayfication over a local ring. Local cohomology annihilators appear through the notion of p-standard system of parameters. For a local ring, we prove an equivalence of the existence of Macaulayfications, the existence of a p-standard system of parameters, being a quotient of a Cohen-Macaulay local ring, and the verification of Faltings’ Annihilator theorem. For a finitely generated module which is unmixed and faithful, we prove an equivalence of the existence of an arithmetic Macaulayfication and the existence of a p-standard system of parameters; and both are proved to be equivalent to the existence of an arithmetic Macaulayfication on the ground ring. A connection between Macaulayfication and universal catenaricity is also discussed.

   37-60
4. Outer-Inner Approximation Projection Methods for Multivalued Variational Inequalities Pham Ngoc Anh , Le Thi Hoai An PDF

   ▸ Abstract

   In this paper, we present new projection methods for solving multivalued variational inequalities on a given nonlinear convex feasible domain. The first one is an extension of the extragradient method to multivalued variational inequalities under the asymptotic optimality condition, but it must satisfy certain Lipschitz continuity conditions. To avoid this requirement, we propose linesearch procedures commonly used in variational inequalities to obtain an approximation linesearch method for solving multivalued variational inequalities. Next, basing on a family of nonempty closed convex subsets of $\mathcal R^{n}$ and linesearch techniques, we give inner approximation projection algorithms for solving multivalued variational inequalities and the convergence of the algorithms is established under few assumptions.

   61-79
5. Capitulation in the Absolutely Abelian Extensions of some Number Fields II Abdelmalek Azizi , Abdelkader Zekhnini , Mohammed Taous PDF

   ▸ Abstract

   We study the capitulation of 2-ideal classes of an infinite family of imaginary biquadratic number fields consisting of fields $\Bbb k =\mathbb{Q}(\sqrt {pq_{1}q_{2}}, i)$, where $i=\sqrt {-1}$ and $q_1\equiv q_2\equiv −p\equiv −1$ (mod 4) are different primes. For each of the three quadratic extensions $\mathbb K/\Bbb k$ inside the absolute genus field $\Bbb k^{(*)}$ of $\Bbb k$, we compute the capitulation kernel of $\mathbb K/\Bbb k$. Then we deduce that each strongly ambiguous class of $\Bbb k/\mathbb Q(i)$ capitulates already in $\Bbb k^{(*)}$.

   81-97
6. Stability Results for Semi-linear Parabolic Equations Backward in Time Nguyen Van Duc , Nguyen Van Thang PDF

   ▸ Abstract

   Let $H$ be a Hilbert space with the norm $\|\cdot\|$, and let $A:D(A) \subset H \to H$ be a positive self-adjoint unbounded linear operator on H such that $−A$ generates a $C_0$ semi-group on $H$. Let $\varphi$ be in $H,$ $E > ε$ a given positive number and let $f : [0, T]\times H \to H$ satisfy the Lipschitz condition $\|f(t, w_1)−f(t, w_2)∥ \leq k\|w_1−w_2\|,$ $w_1,w_2\in H$, for some non-negative constant $k$ independent of $t$, $w_1$ and $w_2$. It is proved that if $u_1$ and $u_2$ are two solutions of the ill-posed semi-linear parabolic equation backward in time $u_t + A u = f(t, u)$, $0 < t \leq T,$ $\|u(T)−\varphi\| \leq \varepsilon$ and $\|u_i(0)\| \leq E$, $i = 1,2,$ then $$\|u_{1}(t)-u_{2}(t)\| \leq 2\varepsilon^{t/T} E^{1-t/T}\exp\Big[\Big(2k+\frac{1}{4}k^{2}(T+t)\Big)\frac{t(T-t)}{T}\Big] \quad \forall t \in [0,T].$$ The ill-posed problem is stabilized by a modification of Tikhonov regularization which yields an error estimate of Hölder type.

   99-111
7. Adaptive Sampling Recovery of Functions with Bounded Modulus of Smoothness Nguyen Manh Cuong , Mai Xuan Thao PDF

   ▸ Abstract

   In this paper, we extend results obtained by Dinh Dũng on optimal methods of adaptive sampling recovery of functions by sets of a finite capacity which is measured by their cardinality or pseudo-dimension, to univariate Besov-type classes of functions with bounded modulus of smoothness.

   113-127
8. Analytical and Numerical Approximation Formulas on the Dunkl-Type Fock Spaces Fethi Soltani , Akram Nemri PDF

   ▸ Abstract

   In this work, we establish some versions of Heisenberg-type uncertainty principles for the Dunkl-type Fock space $F_{k}(\mathbb{C}^{d})$. Next, we give an application of the classical theory of reproducing kernels to the Tikhonov regularization problem for operator $L:F_{k}(\mathbb{C}^{d})\rightarrow H$, where H is a Hilbert space. Finally, we come up with some results regarding the Tikhonov regularization problem and the Heisenberg-type uncertainty principle for the Dunkl-type Segal-Bargmann transform $\mathcal {B}_{k}$. Some numerical applications are given.

   129-147
9. On a Minimal Set of Generators for the Polynomial Algebra of Five Variables as a Module over the Steenrod Algebra Dang Vo Phuc , Nguyen Sum PDF

   ▸ Abstract

   Let $P_k$ be the graded polynomial algebra $\mathbb{F}_{2}[x_{1},x_{2},{\ldots } ,x_{k}]$ over the prime field of two elements, $\mathbb{F}_{2}$, with the degree of each $x_i$ being 1. We study the hit problem, set up by Frank Peterson, of finding a minimal set of generators for $P_k$ as a module over the mod-2 Steenrod algebra, $\mathcal{A}$. In this paper, we explicitly determine a minimal set of $\mathcal{A}$-generators for $P_k$ in the case $k = 5$ and the degree $4(2^d−1)$ with d an arbitrary positive integer.

   149-162
10. On the Evolution Problem for the Einstein-Vlasov System Calvin Tadmon PDF

    ▸ Abstract

    In this work, we solve the evolution problem associated with the Einstein-Vlasov system, with initial data specified on two transversally intersecting null hypersurfaces. The main existence and uniqueness result of the paper is obtained by a contracting mapping principle, combined with Sobolev inequalities and Moser estimates as well as energy inequalities for first order and second order linear hyperbolic systems. The whole investigation is conducted in appropriate weighted Sobolev spaces.

    163-186
11. Unstable Manifolds for Partial Neutral Differential Equations and Admissibility of Function Spaces Thieu Huy Nguyen , Van Bang Pham PDF

    ▸ Abstract

    We prove the existence and attraction property of an unstable manifold for solutions to the partial neutral functional differential equation of the form $$\left\{\begin{array}{ll} \frac{\partial}{\partial t}Fu_{t}= B(t)Fu_{t} +\varPhi(t,u_{t}),\quad t\ge s;~ t,s\in \mathbb{R},\\ u_{s}=\phi\in \mathcal{C}:=C([-r, 0], X) \end{array}\right.$$ under the conditions that the family of linear operators $(B(t))_{t\in \mathbb{R}}$ defined on a Banach space $X$ generates the evolution family $(U(t, s))_{t\geq s}$ having an exponential dichotomy on the whole line $\mathbb R$, the difference operator $F:\mathcal{C}\to X$ is bounded and linear, and the nonlinear delay operator $\Phi$ satisfies the $\varphi$-Lipschitz condition, i.e., $\| \Phi (t,\phi ) -\Phi (t,\psi )\| \le \phi (t)\|\phi -\psi \|_{\mathcal{C}}$ for $\phi ,~ \psi \in \mathcal{C}$, where $\varphi(\cdot)$ belongs to an admissible function space defined on $\mathbb R$. Our main method is based on Lyapunov-Perron’s equations combined with the admissibility of function spaces and the technique of choosing $F$-induced trajectories.

    187-207