Issue 2, Volume 51, June 2026

2026

1. Ulrich Ideals on Rational Triple Points of Dimension Two Kyosuke Maeda , Ken-ichi Yoshida PDF

   ▸ Abstract

   In this paper, we prove that the canonical trace ideal ${\text {trace}}_A(\omega _A)$ is an Ulrich ideal for any two-dimensional rational triple point $A$. Using this, we classify all Ulrich ideals on rational triple points. Moreover, we show that if $(A,{\mathfrak m})$ is a two-dimensional quotient singularity with the multiplicity $e \ge 4$, then ${\mathfrak m}$ is the unique Ulrich ideal of $A$. As a result, we can classify all Ulrich ideals of $A$ if $A$ is either a rational triple point or a quotient singularity.

   161-178
2. A Schmidt’s Subspace Theorem for Moving Hypersurface Targets on Algebraic Varieties Over Function Fields Hoang Duong Tan Hung , Tran Van Tan , Nguyen Van Thin PDF

   ▸ Abstract

   In this paper, we establish a non-effective version of Schmidt’s subspace theorem for moving hypersurface targets in weak general position on algebraic varieties over function fields.

   179-190
3. Modular Invariant Rings and the Weyl Groups of Special Unitary Groups Kenshi Ishiguro , Naito Nishihara PDF

   ▸ Abstract

   For a compact connected Lie group G, the rational cohomology ring of its classifying space BG can be expressed as an invariant ring. Namely, for a maximal torus $T^{n}$ of G, it is well–known that $H^{*}(BG; \mathbb {Q} ) \cong H^{*}(BT^{n}; \mathbb {Q} )^{W(G)}$ which is a polynomial ring. If d divides n, the quotient $SU(n)/\mathbb {Z}_d$ is also a Lie group. Their rational cohomology rings are isomorphic, however, the integral representations of the Weyl groups $W(SU(n)/\mathbb {Z}_d)$ are not equivalent. So we ask if the modular invariant rings under the actions are polynomial. Our main theorem is a generalization of some results in [4] and [12].

   191-200
4. Smoluchowski–Kramers Approximation for the Derivative of Solutions Nguyen Van Tan PDF

   ▸ Abstract

   In this paper, we provide a new property for the Smoluchowski-Kramers approximation of stochastic differential equations. We prove the convergence of the derivative of solutions with respect to the initial condition (when the mass of particles tends to zero). We then use the techniques of Malliavin calculus to obtain an explicit Berry-Esseen error bound for the rate of convergence.

   201-218
5. Dualities for Some Function Spaces of One Variable and Their Applications Le Hai Khoi PDF

   ▸ Abstract

   The present survey paper is devoted to the spaces $A^{-\infty }(\Omega )$, their dual spaces, and their associated applications. Here, the domain $\Omega$ is considered not only as the unit disk $\mathbb {D}$ in the complex plane $\mathbb {C}$, but also as the broader class of the so-called Carathéodory domains in $\mathbb {C}$. Furthermore, this paper provides a detailed consideration of the representation problem, a topic that has remained largely untreated in the literature.

   219-241
6. FlatKnotInfo: The First 1.24 Million Flat Knots Jie Chen PDF

   ▸ Abstract

   We use matchings on Lyndon words to classify flat knots up to 8 crossings. Using flat knots invariants such as the based matrix, the $\phi$-invariant, the flat arrow polynomial, and the flat Jones-Krushkal polynomial, we distinguish all flat knots up to 7 crossings except for five pairs. Among the many flat knots considered, we find examples that are: (i) algebraically slice but not slice; (ii) almost classical (null-homologous) but not slice; (iii) nontrivial but with trivial (primitive) based matrix. The classification data has been curated and is available on FlatKnotInfo, which is an interactive searchable website listing flat knots up to 8 crossings and their invariants. It also provides access to algebraic and diagrammatic information on these knots and is designed to enable users to discover patterns and formulate conjectures on their own.

   243-279
7. On the Transfer of Artinian and Cohen-Macaulay Properties under Module-Finite Extensions Tran Do Minh Chau PDF

   ▸ Abstract

   This paper deals with certain classes of modules under module-finite extensions. Let $\varphi : R\hookrightarrow S$ be a module-finite extension between commutative Noetherian local rings. We investigate the transfer of Artinian module structures and attached primes between $R$ and $S$. We clarify the behavior of local cohomology modules as well as certain structures of finitely generated $S$-modules under the restriction of scalars to $R$ via $\varphi$. We show that $R$ is a quotient of a Cohen-Macaulay local ring if and only if so is $S$. As an application, we characterize the structure of Nagata’s idealization. Using Macaulayfication of algebraic varieties and idealization, we give an example to illustrate the results.

   281-294
8. A Remark on Empirical Measure within Weyl Mean Orbital Pseudo-metric Zhongxuan Yang , Xiaojun Huang PDF

   ▸ Abstract

   In this paper, we investigate the Weyl mean orbital pseudo-metric for Polish dynamical systems and its connections to the properties of the space of invariant measures. We establish equivalent conditions under which the set of invariant measures generated by periodic points is dense in the set of ergodic measures, thereby providing a deeper understanding of the relationship between periodic behavior and measure-theoretic properties in the field of Weyl mean orbital pseudo-metric.

   295-306
9. The Cumulative Distribution Function of a Mixture Model with Normal Error Dang Duc Trong , Nguyen Hoang Thanh , Nguyen Tien Dat , Thai Phuc Hung PDF

   ▸ Abstract

   This paper studies statistical deconvolution, focusing on estimating the cumulative distribution function $F_X$ of a random variable X from observations $Y_j = X_j + \sigma \xi _j$, where $X_j$ and $\xi _j$ are independent copies of $X$ and $\xi$, respectively. We assume that the noise $\xi _j$ is normally distributed with zero mean and unknown variance $\sigma ^2$. Under this framework, we propose a new semi-parametric estimator for $F_X$. Our results show that, over a subclass of Sobolev-type distributions, the estimator achieves a convergence rate comparable to previous studies, demonstrating that the accurate semi-parametric inference for $F_X$ is feasible even when the noise variance is unknown.

   307-336