Authors:Pham Ky Anh; Tran Viet Anh; Le Dung Muu Pages: 413 - 429 Abstract: Abstract In this paper, we investigate a bilevel split variational inequality problem (BSVIP) involving a strongly monotone mapping in the upper-level problem and pseudomonotone mappings in the lower-level one. A strongly convergent algorithm for such a BSVIP is proposed and analyzed. In particular, a problem of finding the minimum-norm solution of a split pseudomonotone variational inequality problem is also studied. As a consequence, we get a strongly convergent algorithm for finding the minimum-norm solution to the split feasibility problem, which requires only two projections at each iteration step. An application to discrete optimal control problems is considered. PubDate: 2017-09-01 DOI: 10.1007/s40306-016-0178-8 Issue No:Vol. 42, No. 3 (2017)

Authors:Dao Quang Khai Pages: 431 - 443 Abstract: Abstract In this paper, we study local well-posedness for the Navier-Stokes equations with arbitrary initial data in homogeneous Sobolev spaces \(\dot {H}^{s}_{p}(\mathbb {R}^{d})\) for \(d \geq 2, p > \frac {d}{2}\) , and \(\frac {d}{p} - 1 \leq s < \frac {d}{2p}\) . The obtained result improves the known ones for p > d and s = 0 (see [4, 6]). In the case of critical indexes \(s=\frac {d}{p}-1\) , we prove global well-posedness for Navier-Stokes equations when the norm of the initial value is small enough. This result is a generalization of the one in [5] in which p = d and s = 0. PubDate: 2017-09-01 DOI: 10.1007/s40306-016-0192-x Issue No:Vol. 42, No. 3 (2017)

Authors:Yan Gu Pages: 445 - 454 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: 2017-09-01 DOI: 10.1007/s40306-017-0204-5 Issue No:Vol. 42, No. 3 (2017)

Authors:Si Duc Quang; Do Phuong An Pages: 455 - 470 Abstract: Abstract Let V be a projective subvariety of \(\mathbb P^{n}(\mathbb C)\) . A family of hypersurfaces \(\{Q_{i}\}_{i=1}^{q}\) in \(\mathbb P^{n}(\mathbb C)\) is said to be in N-subgeneral position with respect to V if for any 1≤i 1<⋯<i N+1≤q, \( V\cap (\bigcap _{j=1}^{N+1}Q_{i_{j}})=\varnothing \) . In this paper, we will prove a second main theorem for meromorphic mappings of \(\mathbb C^{m}\) into V intersecting hypersurfaces in subgeneral position with truncated counting functions. As an application of the above theorem, we give a uniqueness theorem for meromorphic mappings of \(\mathbb C^{m}\) into V sharing a few hypersurfaces without counting multiplicity. In particular, we extend the uniqueness theorem for linearly nondegenerate meromorphic mappings of \(\mathbb C^{m}\) into \(\mathbb P^{n}(\mathbb C)\) sharing 2n+3 hyperplanes in general position to the case where the mappings may be linearly degenerated. PubDate: 2017-09-01 DOI: 10.1007/s40306-016-0196-6 Issue No:Vol. 42, No. 3 (2017)

Authors:M. Aminian; S. M. B. Kashani Pages: 471 - 490 Abstract: Abstract In this paper, we introduce L k -biharmonic hypersurfaces M in simply connected space forms R n+1(c) and propose L k -conjecture for them. For c=0,−1, we prove the conjecture when hypersurface M has two principal curvatures with multiplicities 1,n−1, or M is weakly convex, or M is complete with some constraints on it and on L k . We also show that neither there is any L k -biharmonic hypersurface M n in \( \mathbb {H}^{n+1} \) with two principal curvatures of multiplicities greater than one, nor any L k -biharmonic compact hypersurface M n in \( \mathbb {R}^{n+1} \) or in \( \mathbb {H}^{n+1} \) . As a by-product, we get two useful, important variational formulas. The paper is a sequel to our previous paper, (Taiwan. J. Math., 19, 861–874, 5) in this context. PubDate: 2017-09-01 DOI: 10.1007/s40306-016-0195-7 Issue No:Vol. 42, No. 3 (2017)

Authors:Mehmet Akif Akyol Pages: 491 - 507 Abstract: Abstract As a generalization of semi-invariant submersions, we introduce conformal semi-invariant submersions from almost product Riemannian manifolds onto Riemannian manifolds. We give examples, and investigate the geometry of foliations which arise from the definition of a conformal submersion and show that there are certain product structures on the total space of a conformal semi-invariant submersion. Moreover, we also find necessary and sufficient conditions for a conformal semi-invariant submersion to be totally geodesic. PubDate: 2017-09-01 DOI: 10.1007/s40306-016-0193-9 Issue No:Vol. 42, No. 3 (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:Ha Binh Minh; Chu Binh Minh; Victor Sreeram Abstract: Abstract In this paper, a balanced truncation type of reduction is proposed for unstable continuous-time systems which is based on unstable system reduction originally proposed for discrete systems. This is achieved by first deriving a link between continuous-time and discrete-time systems which is called the extended bilinear mapping. Using this mapping, an unstable continuous-time system reduction method along with its error bounds is then derived. A numerical example is provided to illustrate the effectiveness of the method and a comparison with other relevant methods in the literature is also included. PubDate: 2017-06-23 DOI: 10.1007/s40306-017-0215-2

Authors:Pham Viet Duc; Mai Anh Duc; Pham Nguyen Thu Trang Abstract: Abstract The main goal of this article is to give necessary and sufficient conditions on the tautness modulo an analytic subset of complex spaces. PubDate: 2017-06-23 DOI: 10.1007/s40306-017-0214-3

Authors:Nipen Saikia; Chayanika Boruah Abstract: 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: 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:Si Tiep Dinh; Huy Vui Ha; Tien Son Pham Abstract: 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: 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: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: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