Issue 4, Volume 48, December 2023

2023

1. A Note on Jacobian Problem Over $\mathbb Z$ Nguyen Van Chau PDF

   ▸ Abstract

   Motivated by the Jacobian problem, this article is concerned with the density of the image set $F(\mathbb Z^n)$ of polynomial maps $F\in\mathbb Z[X_1,\dots,X_n]^n$ with $\det DF\equiv 1$. It is shown that if such a map $F$ is not invertible, its image set $F(\mathbb Z^n)$ must be very thin in the lattice $\mathbb Z^n$: (1) for almost all lines $l$ in $\mathbb Z^n$ the numbers $\#(F^{−1}(l)\cap \mathbb Z^n)$ are uniformly bounded; (2) $\#\{z\in F(\mathbb Z^n)\,:\,|z_i|\geq B\} \ll B_{n−1}$ as $B\to +\infty$, where the implicit constants depend on $F$.

   515-521
2. Noetherian Rings of the Form ${\mathcal {A}}[X,Y;\lambda ]$ Abdelamir Dabbabi , Ali Benhissi PDF

   ▸ Abstract

   In this paper, we use composite ring extensions to construct a new class of Noetherian rings. Composite ring extensions are examples of pullback constructions, and they are useful in constructing of (counter)-examples.

   523-531
3. On Leavitt Path Algebras of Hopf Graphs T. G. Nam , N. T. Phuc PDF

   ▸ Abstract

   In this paper, we provide the structure of Hopf graphs associated to pairs $(G,\mathfrak {r})$ consisting of groups $G$ together with ramification datas $\mathfrak {r}$ and their Leavitt path algebras. Consequently, we characterize the Gelfand-Kirillov dimension, the stable rank, the purely infinite simplicity and the existence of a nonzero finite dimensional representation of the Leavitt path algebra of a Hopf graph via properties of ramification data $\mathfrak {r}$ and $G$.

   533-549
4. On the Numerical Solution to an Inverse Medium Scattering Problem Dinh-Liem Nguyen , Trung Truong PDF

   ▸ Abstract

   This paper is concerned with the inverse medium scattering problem of determining the location and shape of penetrable scattering objects from measurements of the scattered field. We study a sampling indicator function for recovering the scattering object in a fast and robust way. A flexibility of this indicator function is that it is applicable to data measured in near-field regime or far-field regime. The implementation of the function is simple and does not involve solving any ill-posed problems. The resolution analysis and stability estimate of the indicator function are investigated using the factorization analysis of the far-field operator along with the Funk-Hecke formula. The performance of the method is verified on both simulated and experimental data.

   551-566
5. The Tor Algebra of Trimmings of Gorenstein Ideals Luigi Ferraro , Alexis Hardesty PDF

   ▸ Abstract

   Let $(R,\mathfrak m,\Bbbk )$ be a regular local ring of dimension 3. Let $I$ be a Gorenstein ideal of $R$ of grade 3. Buchsbaum and Eisenbud proved that there is a skew-symmetric matrix of odd size such that $I$ is generated by the sub-maximal pfaffians of this matrix. Let $J$ be the ideal obtained by multiplying some of the pfaffian generators of $I$ by $\mathfrak m$; we say that $J$ is a trimming of $I$. Building on a recent paper of Vandebogert, we construct an explicit free resolution of $R/J$ and compute a partial DG algebra structure on this resolution. We provide the full DG algebra structure in the appendix. We use the products on this resolution to study the Tor algebra of such trimmed ideals and we use the information obtained to prove that recent conjectures of Christensen, Veliche and Weyman on ideals of class $\textbf{G}$ hold true in our context. Furthermore, we address the realizability question for ideals of class $\textbf{G}$.

   567-604
6. Non-normal Edge Rings Satisfying $(S_{2})$-condition Nayana Shibu Deepthi PDF

   ▸ Abstract

   Let $G$ be a finite simple connected graph on the vertex set $V(G)=[d]=\{1,\dots ,d\}$ with edge set $E(G)=\{e_{1},\dots , e_{n}\}$. Let $\mathbb {K}[\textbf{t}]=\mathbb {K}[t_{1},\dots ,t_{d}]$ be the polynomial ring in d variables over a field $\mathbb {K}$. The edge ring of G is the affine semigroup ring $\mathbb {K}[G]$ generated by monomials $\textbf{t}^{e}:=t_{i}t_{j}$, for $e=\{i,j\} \in E(G)$. In this paper, we will prove that, given integers $d$ and $n$, where $d\geq 7$ and $d+1\le n\le \frac{d^{2}-7d+24}{2}$, there exists a finite simple connected graph $G$ with $|V(G)|=d$ and $|E(G)|=n$, such that $\mathbb {K}[G]$ is non-normal and satisfies $(S_{2})$-condition.

   605-619
7. An Inertial Iterative Algorithm for Approximating Common Solutions to Split Equalities of Some Nonlinear Optimization Problems O. T. Mewomo , G. N. Ogwo , T. O. Alakoya PDF

   ▸ Abstract

   In this paper, we introduce a new inertial Tseng’s extragradient method with self-adaptive step sizes for approximating a common solution of split equalities of equilibrium problem (EP), non-Lipschitz pseudomonotone variational inequality problem (VIP) and fixed point problem (FPP) of nonexpansive semigroups in real Hilbert spaces. We prove that the sequence generated by our proposed method converges strongly to a common solution of the EP, pseudomonotone VIP and FPP of nonexpansive semigroups without any linesearch procedure nor the sequential weak continuity condition often assumed by authors when solving non-Lipschitz VIPs. Finally, we provide some numerical experiments for the proposed method in comparison with related methods in the literature. Our result improves, extends and generalizes several of the existing results in this direction.

   621-650
8. Some Types of Carathéodory Scheme for Caputo Stochastic Fractional Differential Equations in $L^p$ Spaces Phan Thi Huong , Pham The Anh PDF

   ▸ Abstract

   In this paper, we construct Carathéodory type and exponential Carathéodory type schemes for Caputo stochastic fractional differential equations (CSFDEs) of order $\alpha \in (\frac{1}{2},1)$ in $L^p$ spaces with $p \geq 2$ whose coefficients satisfy a standard Lipschitz and a linear growth bound conditions. The strong convergence and the convergence rate of these schemes are also established.

   651-669
9. On Some Operator Inequalities with Respect to the $s$-Convexity Lahcen Tarik , Mustapha Raïssouli , Mohamed Chergui , Bouazza El Wahbi PDF

   ▸ Abstract

   In this paper, we extend the concept of s-convexity from the case where the functions are with real variables to the case where the functions are with operator arguments. Afterwards, we investigate some related properties and operator inequalities. As an application, some inequalities of Hermite-Hadamard and Jensen types involving some operator means are established.

   671-690
10. Differential Stability in a Multi-Objective Optimal Control Problems with a Possibly Empty Solution Set N. T. Toan , L. Q. Thuy PDF

    ▸ Abstract

    This paper studies the first-order behavior of the weak optimal value mapping in a parametric multi-objective discrete optimal control problem under linear state equations, where the solution set may be empty. By establishing an abstract result on the $\varepsilon$-weak subdifferential of the weak optimal value mapping in a parametric multi-objective mathematical programming problem with an inclusion constraint, we derive a formula for computing the $\varepsilon$-weak subdifferential of the weak optimal value mapping in a parametric multi-objective discrete optimal control problem. The obtained results are proved directly without using scalarization techniques.

    691-707