Authors:Tran Van Nghi; Nguyen Nang Tam Pages: 311 - 336 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: 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: 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: 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: 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:Pham Viet Hung Abstract: We investigate the rate of convergence for the central limit theorems of sojourn times on the growing sphere of isotropic Gaussian fields defined on the whole Euclidean space. In the case of the sojourn times defined on a cube, the similar problem has been studied by using the Malliavin-Stein method. Following this idea, in this paper, we establish the explicit rate for various probability distances with a careful examination of the variance. PubDate: 2017-05-19 DOI: 10.1007/s40306-017-0212-5

Authors:Nguyen Minh Tri; Tran Tuan Nam Abstract: We introduce the ideal transform functor D I,J with respect to a pair of ideals (I,J) which is an extension of the ideal transform functor D I of Brodmann. Some equivalent conditions on the exactness of the ideal transform functor will be shown in the paper. We also study the finiteness of the sets Ass R (R i D I (N)) and Ass R (R i D I, J (M,N)). PubDate: 2017-05-17 DOI: 10.1007/s40306-017-0213-4

Authors:Nipen Saikia; Chayanika Boruah Abstract: For any positive integer ℓ, let B ℓ (n) denotes the number of ℓ-regular partition triples of a positive integer n. By employing q −series identities, we prove infinite family of arithmetic identities and congruences modulo 4 for B 2(n), modulo 2 and 9 for B 3(n), modulo 2 for B 4(n) and modulo 2 and 5 for B 5(n). PubDate: 2017-04-25 DOI: 10.1007/s40306-017-0206-3

Authors:Edoardo Ballico Abstract: We give the stratification by the symmetric tensor rank of all degree d ≥ 9 homogeneous polynomials with border rank 5 and which depend essentially on at least five variables, extending previous works (A. Bernardi, A. Gimigliano, M. Idà, E. Ballico) on lower border ranks. For the polynomials which depend on at least five variables, only five ranks are possible: 5, d + 3, 2d + 1, 3d − 1, 4d − 3, but each of the ranks 3d − 1 and 2d + 1 is achieved in two geometrically different situations. These ranks are uniquely determined by a certain degree 5 zero-dimensional scheme A associated with the polynomial. The polynomial f depends essentially on at least five variables if and only if A is linearly independent (in all cases, f essentially depends on exactly five variables). The polynomial has rank 4d − 3 (resp. 3d − 1, resp. 2d + 1, resp. d + 3, resp. 5) if A has 1 (resp. 2, resp. 3, resp. 4, resp. 5) connected component. The assumption d ≥ 9 guarantees that each polynomial has a uniquely determined associated scheme A. In each case, we describe the dimension of the families of the polynomials with prescribed rank, each irreducible family being determined by the degrees of the connected components of the associated scheme A. PubDate: 2017-04-22 DOI: 10.1007/s40306-017-0211-6

Authors:Hoang Viet Long; Nguyen Thi Kim Son; Ha Thi Thanh Tam; Jen-Chih Yao Abstract: In this paper, the solvability of Darboux problems for nonlinear fractional partial integro-differential equations with uncertainty under Caputo gH-fractional differentiability is studied in the infinity domain J ∞ = [0,∞) × [0,∞). New concepts of Hyers-Ulam stability and Hyers-Ulam-Rassias stability for these problems are also investigated through the equivalent integral forms. A computational example is presented to demonstrate our main results. PubDate: 2017-04-22 DOI: 10.1007/s40306-017-0207-2

Authors:Si Tiep Dinh; Huy Vui Ha; Tien Son Pham Abstract: Let F := (f 1, …, f p ): ℝ n → ℝ p be a polynomial map, and suppose that S := {x ∈ ℝ n : f i (x) ≤ 0,i = 1, …, p}≠∅. Let d := maxi =1, …, p deg f i and \(\mathcal {H}(d, n, p) := d(6d - 3)^{n + p - 1}.\) Under the assumptions that the map F : ℝ n → ℝ p is convenient and non-degenerate at infinity, we show that there exists a constant c > 0 such that the following so-called Hölder-type global error bound result holds \(c d(x,S) \le [f(x)]_{+}^{\frac {2}{\mathcal {H}(2d, n, p)}} + [f(x)]_{+} \quad \textrm { for all } \quad x \in \mathbb {R}^{n},\) where d(x,S) denotes the Euclidean distance between x and S, f(x) := maxi=1, …, p f i (x), and [f(x)]+ := max{f(x),0}. The class of polynomial maps (with fixed Newton polyhedra), which are non-degenerate at infinity, is generic in the sense that it is an open and dense semi-algebraic set. Therefore, Hölder-type global error bounds hold for a large class of polynomial maps, which can be recognized relatively easily from their combinatoric data. This follows up the result on a Frank-Wolfe type theorem for non-degenerate polynomial programs in Dinh et al. (Mathematical Programming Series A, 147(16), 519–538, 2014). PubDate: 2017-04-21 DOI: 10.1007/s40306-017-0209-0

Authors:Mohamed Zitane 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 DOI: 10.1007/s40306-017-0208-1

Authors:Ramu Geddavalasa; P. Sam Johnson 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: 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:Kyoji Saito 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: 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: 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: 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: 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: 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