Issue 4, Volume 44, December 2019

2019

1. Combinatorial Algorithms for the Uniform-Cost Inverse 1-Center Problem on Weighted Trees Kien Trung Nguyen , Huong Nguyen-Thu , Nguyen Thanh Hung PDF

   ▸ Abstract

   Inverse 1-center problem on a network is to modify the edge lengths or vertex weights within certain bounds so that the prespecified vertex becomes an (absolute) 1-center of the perturbed network and the modifying cost is minimized. This paper focuses on the inverse 1-center problem on a weighted tree with uniform cost of edge length modification, a generalization for the analogous problem on an unweighted tree (Alizadeh and Burkard, Discrete Appl. Math. 159, 706–716, 2011). To solve this problem, we first deal with the weighted distance reduction problem on a weighted tree. Then, the weighted distances balancing problem on two rooted trees is introduced and efficiently solved. Combining these two problems, we derive a combinatorial algorithm with complexity of $O(n^{2})$ to solve the inverse 1-center problem on a weighted tree if there exists no topology change during the edge length modification. Here, n is the number of vertices in the tree. Dropping this condition, the problem is solvable in $O(n^{2}\mathbf {c})$ time, where $\mathbf {c}$ is the compressed depth of the tree. Finally, some special cases of the problem with improved complexity, say linear time, are also discussed.

   813-831
2. On Flexible Sequences Bruno Dinis , Nam Van Tran , Imme van den Berg PDF

   ▸ Abstract

   In the setting of nonstandard analysis, we introduce the notion of flexible sequence. The terms of flexible sequences are external numbers. These are a sort of analogue for the classical $O(\cdot)$ and $o(\cdot)$ notation for functions, and have algebraic properties similar to those of real numbers. The flexibility originates from the fact that external numbers are stable under some shifts, additions, and multiplications. We introduce two forms of convergence and study their relation. We show that the usual properties of convergence of sequences hold or can be adapted to these new notions of convergence and give some applications.

   833-874
3. Asymptotic Behavior of Solutions to Semilinear Parabolic Equations with Infinite Delay Dang Thi Phuong Thanh PDF

   ▸ Abstract

   We consider a class of semilinear parabolic equations with infinite delay and nonlinearity of polynomial type. We first prove the existence and uniqueness of weak solutions by using the Galerkin method. Then, we show the existence of a compact global attractor for the continuous semigroup associated to the problem. The existence and exponential stability of weak stationary solutions are also investigated.

   875-892
4. Non-uniform Berry-Esseen Bounds for Coordinate Symmetric Random Vectors with Applications Le Van Thanh , Nguyen Ngoc Tu PDF

   ▸ Abstract

   This work establishes the nonuniform Berry-Esseen inequality for coordinate symmetric vectors. The nonuniform $L_p$ $(p \geq  1)$ bound is also established. The main results are applied to projections of random vectors distributed according to a family of measures on the ${\ell _{r}^{n}}$ sphere and the ${\ell _{r}^{n}}$ ball, including cone measure and volume measure.

   893-904
5. On the Annihilator Submodules and the Annihilator Essential Graph Sakineh Babaei , Shiroyeh Payrovi , Esra Sengelen Sevim PDF

   ▸ Abstract

   Let $R$ be a commutative ring and let $M$ be an $R$-module. For $a \in R$, $\mathrm{Ann}_M(a) = \{m \in M \,:\, am = 0\}$ is said to be an annihilator submodule of $M$. In this paper, we study the property of being prime or essential for annihilator submodules of $M$. Also, we introduce the annihilator essential graph of equivalence classes of zero divisors of $M$, $\mathrm{AE}_R(M)$, which is constructed from classes of zero divisors, determined by annihilator submodules of $M$ and distinct vertices $[a]$ and $[b]$ are adjacent whenever $\mathrm{Ann}_M(a) + \mathrm{Ann}M(b)$ is an essential submodule of $M$. Among other things, we determine when $\mathrm{AE}_R(M)$ is a connected graph, a star graph, or a complete graph. We compare the clique number of $\mathrm{AE}_R(M)$ and the cardinal of $-\mathrm{Ass}_R(M)$.

   905-914
6. Volume of Sublevel Sets Versus Area of Level Sets via Gelfand-Leray Form Duc Tai Trinh PDF

   ▸ Abstract

   In this paper, we give a relation between the volume of sublevel sets and the area of level sets using a Gelfand-Leray form. As a consequence, we give an estimation of the volume of sublevel sets. In particular, we give a proof of the known fact that the derivative of the volume of a $n$-dimensional ball with respect to the radius equals the area of the sphere which bounds the $n$-dimensional ball.

   915-922
7. On Properties of Expansive Group Actions Ali Barzanouni , Mahin Sadat Divandar , Ekta Shah PDF

   ▸ Abstract

   We study several properties of expansive group actions on metric spaces and obtain some relations between the expansivity for subgroup and group actions. Through counter examples, the necessity of hypothesis is justified. We also study the expansivity of covering maps. Further, we define orbit expansivity for group actions on topological spaces and use it to characterize expansive actions.

   923-934
8. Two Strong Convergence Theorems for the Common Null Point Problem in Banach Spaces Truong Minh Tuyen , Nguyen Minh Trang PDF

   ▸ Abstract

   In this paper, we study the common null point problem in Banach spaces. Then, using the shrinking projection method and $\varepsilon$-enlargement of maximal monotone operator, we prove two strong convergence theorems with nonsummable errors for solving this problem.

   935-953
9. Stable Numerical Solution for a Class of Structured Differential-Algebraic Equations by Linear Multistep Methods Vu Hoang Linh , Nguyen Duy Truong PDF

   ▸ Abstract

   It is known that when we apply a linear multistep method to differential-algebraic equations (DAEs), usually the strict stability of the second characteristic polynomial is required for the zero stability. In this paper, we revisit the use of linear multistep discretizations for a class of structured strangeness-free DAEs. Both explicit and implicit linear multistep schemes can be used as underlying methods. When being applied to an appropriately reformulated form of the DAEs, the methods have the same convergent order and the same stability property as applied to ordinary differential equations (ODEs). In addition, the strict stability of the second characteristic polynomial is no longer required. In particular, for a class of semi-linear DAEs, if the underlying linear multistep method is explicit, then the computational cost may be significantly reduced. Numerical experiments are given to confirm the advantages of the new discretization schemes.

   955-976
10. Differential Extensions of Weakly Principally Quasi-Baer Rings Kamal Paykan , Ahmad Moussavi PDF

    ▸ Abstract

    A ring $R$ is called weakly principally quasi-Baer or simply (weakly p.q.-Baer) if the right annihilator of a principal right ideal is right $s$-unital by right semicentral idempotents, which implies that $R$ modulo, the right annihilator of any principal right ideal, is flat. We study the relationship between the weakly p.q.-Baer property of a ring $R$ and those of the differential polynomial extension $R[x;\delta]$, the pseudo-differential operator ring $R((x^{− 1};\delta))$, and also the differential inverse power series extension $R[[x^{− 1};\delta]]$ for any derivation $\delta$ of $R$. Examples to illustrate and delimit the theory are provided.

    977-991
11. Upper Triangular Operator Matrices, SVEP, and Property $(w)$ Mohammad H. M. Rashid PDF

    ▸ Abstract

    When $A\in \mathscr{L}(\mathbb {X})$ and $B\in \mathscr{L}(\mathbb {Y})$ are given, we denote by $M_C$ an operator acting on the Banach space $\mathbb {X}\oplus \mathbb {Y}$ of the form $M_{C}=\left (\begin {array}{cccccccc} A & C \\ 0 & B \\ \end {array}\right )$. In this paper, first we prove that $\sigma_w(M_0) = \sigma_w(M_C)\cup\{S(A^∗) \cap S(B)\}$ and $\mathbf {\sigma}_{aw}(M_{C})\subseteq \mathbf {\sigma}_{aw}(M_{0})\cup S_{+}^{*}(A)\cup S_{+}(B)$. Also, we give the necessary and sufficient condition for $M_C$ to be obeys property $(w)$. Moreover, we explore how property $(w)$ survive for $2 \times 2$ upper triangular operator matrices $M_C$. In fact, we prove that if $A$ is polaroid on $E^{0}(M_{C})=\{\lambda \in \mathrm{iso\,}\sigma (M_{C}):0<\dim (M_{C}-\lambda )^{-1}\}$, $M_0$ satisfies property $(w)$, and $A$ and $B$ satisfy either the hypotheses (i) $A$ has SVEP at points $\mathbf {\lambda }\in \mathbf {\sigma }_{aw}(M_{0})\setminus \mathbf {\sigma }_{SF_{+}}(A)$ and $A^∗$ has SVEP at points $\mu \in \mathbf {\sigma }_{w}(M_{0})\setminus \mathbf {\sigma }_{SF_{+}}(A)$, or (ii) $A^∗$ has SVEP at points $\mathbf {\lambda }\in \mathbf {\sigma }_{w}(M_{0})\setminus \mathbf {\sigma }_{SF_{+}}(A)$ and $B^∗$ has SVEP at points $\mu \in \mathbf {\sigma }_{w}(M_{0})\setminus \mathbf {\sigma }_{SF_{+}}(B)$, then $M_C$ satisfies property $(w)$. Here, the hypothesis that points $λ \in E^0(M_C)$ are poles of $A$ is essential. We prove also that if $S(A^∗) \cup S(B^∗)$, points $\mathbf {\lambda }\in {E_{a}^{0}}(M_{C})$ are poles of $A$ and points $\mu \in {E_{a}^{0}}(B)$ are poles of $B$, then $M_C$ satisfies property $(w)$. Also, we give an example to illustrate our results.

    993-1004