Authors:Tran Van Nghi; Nguyen Nang Tam Pages: 311 - 336 Abstract: Abstract The aim of this paper is to investigate the continuity and the directional differentiability of the value function in quadratically constrained nonconvex quadratic programming problem. Our result can be used in some cases where the existing results on differential stability in nonlinear programming (applied to quadratic programming) cannot be used. PubDate: 2017-06-01 DOI: 10.1007/s40306-016-0179-7 Issue No:Vol. 42, No. 2 (2017)

Authors:Tran Van Thang Pages: 337 - 355 Abstract: Abstract In this article, we present a conjugate duality for nonconvex optimization problems. This duality scheme is symmetric and has zero gap. As applied to a vector-maximization problem, it transforms the latter into an optimization problem over a weakly efficient set which can be solved by monotonic optimization methods. PubDate: 2017-06-01 DOI: 10.1007/s40306-016-0182-z Issue No:Vol. 42, No. 2 (2017)

Authors:Dao Trong Quyet; Nguyen Viet Tuan Pages: 357 - 367 Abstract: Abstract We consider g-Navier-Stokes equations in a two-dimensional smooth bounded domain Ω. First, we study the existence and exponential stability of a stationary solution under some certain conditions. Second, we prove that any unstable steady state can be stabilized by proportional controller with support in an open subset \(\omega \subset {\Omega }\) such that Ω∖ω is sufficiently “small.” PubDate: 2017-06-01 DOI: 10.1007/s40306-016-0180-1 Issue No:Vol. 42, No. 2 (2017)

Authors:Tran Dinh Phung Pages: 369 - 394 Abstract: Abstract We prove some weighted inequalities for compositions of functions on time scales which are in turn applied to establish some new dynamic Opial-type inequalities in several variables. Some generalizations and applications to partial differential dynamic equations are also considered. PubDate: 2017-06-01 DOI: 10.1007/s40306-016-0187-7 Issue No:Vol. 42, No. 2 (2017)

Authors:Nguyen Van Ngoc Pages: 395 - 411 Abstract: Abstract The aim of the present work is to consider a mixed boundary value problem for the biharmonic equation in a strip. The problem may be interpreted as a deflection surface of a strip plate with the edges y=0,y = h having clamped conditions on intervals x ≥a and hinged support conditions for x <a. Using the Fourier transform, the problem is reduced to studying a system of dual integral equations on the edges of the strip. The uniqueness and existence theorems of solution of system of dual integral equations are established in appropriate Sobolev spaces. A method for reducing the dual integral equation to infinite system of linear algebraic equations is also proposed. PubDate: 2017-06-01 DOI: 10.1007/s40306-016-0191-y Issue No:Vol. 42, No. 2 (2017)

Authors:Ta Van Tu Pages: 1 - 25 Abstract: 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. PubDate: 2017-03-01 DOI: 10.1007/s40306-015-0164-6 Issue No:Vol. 42, No. 1 (2017)

Authors:Razieh Moradi; Mahdieh Ebrahimpour Pages: 27 - 35 Abstract: 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 ϕ-2-absorbing primary submodule if r s x∈N∖ϕ(N) implies r x∈N, or s x∈N, or \(rs\in \sqrt {(N:M)}\) , where r,s∈R and x∈M. In this paper, we study ϕ-2-absorbing primary submodules and we prove some basic properties of these submodules. Also, we give a characterization of ϕ-2-absorbing primary submodules and we investigate ϕ-2-absorbing primary submodules of some well-known modules. PubDate: 2017-03-01 DOI: 10.1007/s40306-016-0181-0 Issue No:Vol. 42, No. 1 (2017)

Authors:Dang Vo Phuc; Nguyen Sum Pages: 149 - 162 Abstract: 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. PubDate: 2017-03-01 DOI: 10.1007/s40306-016-0190-z Issue No:Vol. 42, No. 1 (2017)

Authors:Thieu Huy Nguyen; Van Bang Pham Pages: 187 - 207 Abstract: 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≥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 Φ satisfies the ϕ-Lipschitz condition, i.e., \(\ \Phi (t,\phi ) -\Phi (t,\psi )\ \le \phi (t)\ \phi -\psi \ _{\mathcal {C}}\) for \(\phi ,~ \psi \in \mathcal {C}\) , where ϕ(⋅) 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. PubDate: 2017-03-01 DOI: 10.1007/s40306-016-0183-y Issue No:Vol. 42, No. 1 (2017)

Abstract: Abstract In this work, we study the existence of periodic solutions for some non-autonomous nonlinear partial functional differential equation of neutral type. We assume that the linear part is non-densely defined and generates an evolution family under the conditions introduced by N. Tanaka. The delayed part is assumed to be ω-periodic with respect to the first argument. Using a fixed-point theorem for multivalued mapping, some sufficient conditions are given to prove the existence of periodic solutions. An example is shown to illustrate our results. PubDate: 2017-04-17

Authors:Ramu Geddavalasa; P. Sam Johnson Abstract: Abstract A family of local atoms in a Banach space has been introduced and it has been generalized to an atomic system for operators in Banach spaces, which has been further led to introduce new frames for operators by Dastourian and Janfada, by making use of semi-inner products. Unlike the traditional way of considering sequences in the dual space, sequences in the original space are considered to study them. Appropriate changes have been made in the definitions of atomic systems and frames for operators to fit them for sequences in the dual space without using semi-inner products so that the new notion for Banach spaces can be thought of as a generalization of Banach frames. With some crucial assumptions, we show that frames for operators in Banach spaces share nice properties of frames for operators in Hilbert spaces. PubDate: 2017-04-07 DOI: 10.1007/s40306-017-0210-7

Authors:Yan Gu Abstract: Abstract Let \(\tilde {C_{n}}\) be the graph by adding an ear to C n and \(I=I(\tilde {C_{n}})\) be its edge ideal. In this paper, we prove that \(\operatorname {reg}(I^{s})=2s+\lfloor \frac {n+1}{3}\rfloor -1\) for all s ≥ 1. Let G be the bicyclic graph C m ⊔ C n with edge ideal I = I(G); we compute the regularity of I s for all s ≥ 1. In particular, in some cases, we get \(\operatorname {reg}(I^{s})=2s+\lfloor \frac {m}{3}\rfloor +\lfloor \frac {n}{3}\rfloor -1\) for all s ≥ 2. PubDate: 2017-03-30 DOI: 10.1007/s40306-017-0204-5

Authors:Majid Kowkabi; Behrooz Mashayekhy; Hamid Torabi Abstract: Abstract In this paper, by reviewing the concept of semicovering maps, we present some conditions under which a local homeomorphism becomes a semicovering map. We also obtain some conditions under which a local homeomorphism is a covering map. PubDate: 2017-03-30 DOI: 10.1007/s40306-017-0205-4

Authors:Gholamreza Pirmohammadi; Khadijeh Ahmadi Amoli; Kamal Bahmanpour Abstract: Abstract Let \((R,\operatorname {\frak m})\) be a commutative Noetherian local ring. In this paper, it is shown that the going-up theorem holds for \(R\subseteq \widehat {R}\) if and only if \(\operatorname {Rad}(I+\operatorname {Ann}_{R} A)=\operatorname {\frak m}\) for any proper ideal I of R and any non-zero Artinian I-cofinite module A. Furthermore, using the main result of Zöschinger, Arch. Math. 95, 225–231 (2010), it is shown that these equivalent conditions are equivalent to R being formal catenary with α(R) = 0 and to \(\operatorname {Att}_{R} H^{\dim M}_{I}(M)=\{\operatorname {\frak p} \in \operatorname {Assh}_{R}(M)\,:\,\operatorname {Rad}(\operatorname {\frak p}+I)=\operatorname {\frak m}\}\) for any ideal I of R and any non-zero finitely generated R-module M. PubDate: 2017-03-21 DOI: 10.1007/s40306-017-0203-6

Authors:Kyoji Saito Abstract: Abstract We are interested in the zero locus of a Chapoton’s F-triangle as a polynomial in two real variables x and y. An expectation is that (1) the F-triangle of rank l as a polynomial in x for each fixed y∈[0,1]has exactly l distinct real roots in [0,1], and (2) ith root x i (y) (1≤i≤l) as a function on y∈[0,1]is monotone decreasing. In order to understand these phenomena, we slightly generalized the concept of F-triangles and study the problem on the space of such generalized triangles. We analyze the case of low rank in details and show that the above expectation is true. We formulate inductive conjectures and questions for further rank cases. This study gives a new insight on the zero loci of f +- and f-polynomials. PubDate: 2017-03-14 DOI: 10.1007/s40306-017-0202-7

Authors:Kazumasa Inaba; Masaharu Ishikawa; Masayuki Kawashima; Nguyen Tat Thang Abstract: Abstract We will show that for each k≠1, there exists an isolated singularity of a real analytic map from \(\mathbb {R}^{4}\) to \(\mathbb {R}^{2}\) which admits a real analytic deformation such that the set of singular values of the deformed map has a simple, innermost component with k outward cusps and no inward cusps. Conversely, such a singularity does not exist if k=1. PubDate: 2017-01-20 DOI: 10.1007/s40306-016-0200-1

Authors:Tadashi Ishibe Abstract: Abstract Let Φ be an irreducible (possibly noncrystallographic) root system of rank l of type P. For the corresponding cluster complex Δ(P), which is known as pure (l − 1)-dimensional simplicial complex, we define the generating function of the number of faces of Δ(P) with dimension i − 1, which is called f-polynomial. We show that the f-polynomial has exactly l simple real zeros on the interval (0, 1) and the smallest root for the infinite series of type A l , B l , and D l monotone decreasingly converges to zero as the rank l tends to infinity. We also consider the generating function (called the f +-polynomial) of the number of faces of the positive part Δ+(P) of the complex Δ(P) with dimension i − 1, whose zeros are real and simple and are located in the interval (0, 1), including a simple root at t = 1. We show that the roots in decreasing order of f-polynomial alternate with the roots in decreasing order of f +-polynomial. PubDate: 2017-01-19 DOI: 10.1007/s40306-016-0201-0

Authors:Nguyen Viet Dung; Nguyen Van Ninh Abstract: Abstract Let ð“ be a fiber type arrangement of hyperplanes in ℂ n with complement M(ð“) (see Orlik, O., Terao, H. 1992). In this paper, we will give an explicit formula for the higher topological complexity T C n for the complement M(ð“) in terms of exponents of the arrangement ð“. PubDate: 2017-01-12 DOI: 10.1007/s40306-016-0199-3

Authors:Katsusuke Nabeshima; Shinichi Tajima Abstract: Abstract Complex analytic invariants of hypersurface isolated singularities are considered in the context of symbolic computation. The motivations for this paper are computer calculations of μ ∗-sequences that introduced by B. Teissier to study the Whitney equisingularity of deformations of complex hypersurfaces. A new algorithm that utilizes parametric local cohomology systems is proposed to compute μ ∗-sequences. Lists of μ ∗-sequences of some typical cases are also given. PubDate: 2016-12-19 DOI: 10.1007/s40306-016-0198-4

Authors:Lê Quy Thuong Abstract: Abstract In Kontsevich-Soibelman’s theory of motivic Donaldson-Thomas invariants for 3-dimensional noncommutative Calabi-Yau varieties, the integral identity conjecture plays a crucial role as it involves the existence of these invariants. A purpose of this note is to show how the conjecture arises. Because of the integral identity’s nature, we shall give a quick tour on theories of motivic integration, which lead to a proof of the conjecture for algebraically closed ground fields of characteristic zero. PubDate: 2016-12-19 DOI: 10.1007/s40306-016-0197-5