Issue 3, Volume 42, September 2017

2017

1. On Bilevel Split Pseudomonotone Variational Inequality Problems with Applications Pham Ky Anh , Tran Viet Anh , Le Dung Muu PDF

   ▸ 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.

   413-429
2. Well-Posedness for the Navier-Stokes Equations with Datum in the Sobolev Spaces Dao Quang Khai PDF

   ▸ 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 $s = 0$.

   431-443
3. Regularity of Powers of Edge Ideals of Some Graphs Yan Gu PDF

   ▸ 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 $s \geq 1$. Let $G$ be the bicyclic graph $C_m \sqcup C_n$ with edge ideal $I = I(G)$; we compute the regularity of $I^s$ for all $s \geq 1$. 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 $s \geq 2$.

   445-454
4. Second Main Theorem and Unicity of Meromorphic Mappings for Hypersurfaces in Projective Varieties Si Duc Quang , Do Phuong An PDF

   ▸ 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 $1\leq i_1 < \dots < i_{N+1}\leq q$, $V\cap (\bigcap _{j=1}^{N+1}Q_{i_{j}})=\varnothing$. 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.

   455-470
5. $L_k$-biharmonic Hypersurfaces in Space Forms M. Aminian , S. M. B. Kashani PDF

   ▸ 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.

   471-490
6. Conformal Semi-Invariant Submersions from Almost Product Riemannian Manifolds Mehmet Akif Akyol PDF

   ▸ 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.

   491-507
7. The Stratification by Rank for Homogeneous Polynomials with Border Rank 5 which Essentially Depend on Five Variables Edoardo Ballico PDF

   ▸ Abstract

   We give the stratification by the symmetric tensor rank of all degree $d \geq 9$ 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 $d \geq 9$ 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 $A$.

   509-531
8. Periodic Solutions for Non-Autonomous Neutral Functional Differential Equations with Finite Delay Mohamed Zitane PDF

   ▸ 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 $\omega$-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.

   533-550
9. Congruences of $\ell$− Regular Partition Triples for $\ell\in \{2, 3, 4, 5\}$ Nipen Saikia , Chayanika Boruah PDF

   ▸ Abstract

   For any positive integer $\ell$, let $B_{\ell}(n)$ denotes the number of $\ell$-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 $B_5(n)$.

   551-561
10. Hölder-Type Global Error Bounds for Non-degenerate Polynomial Systems Si Tiep Dinh , Huy Vui Ha , Tien Son Pham PDF

    ▸ Abstract

    Let $F := (f_1, \dots, f_p ): \mathbb R^n tp \mathbb R^p$ be a polynomial map, and suppose that $S := \{x \in \mathbb R^n : f_i(x) \geq 0,i = 1, \dots, p\}\not = \emptyset$. Let $d := \max_{i =1, \dots, p} \deg f_i$ and $\mathcal {H}(d, n, p) := d(6d - 3)^{n + p - 1}.$ Under the assumptions that the map $F : \mathbb R^n \to \mathbb R^p$ 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 $S$, $f(x) := \max_{i=1, \dots, p} f_i(x)$, 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).

    563-585