Issue 3, Volume 51, September 2026

2026

1. Problems on the Conductor of Finite Group Characters Nguyen N. Hung PDF

   ▸ Abstract

   This paper reviews recent results and open problems on the conductor of finite group characters, highlighting their connections to one another and to broader topics in the representation theory of finite groups.

   337-351
2. Endomorphism Algebras Over Commutative Rings and Torsion in Self Tensor Products Justin Lyle PDF

   ▸ Abstract

   Let $R$ be a commutative Noetherian local ring. We study tensor products involving a finitely generated $R$-module $M$ through the natural action of its endomorphism ring. In particular, we study torsion properties of self tensor products in the case where ${\text {End}}_R(M)$ has an $R^*$-algebra structure, and prove that if $M$ is indecomposable, then $M \otimes _{{\text {End}}_R(M)} M$ must always have torsion in this case under mild hypotheses.

   353-361
3. A Note on Cartier Duals of Certain Finite Group Schemes of Prime Power Order Michio Amano PDF

   ▸ Abstract

   Let $\mathcal {G}^{(\lambda )}$ be a group scheme that specializes to ${\mathbb {G}}_a$ when $\lambda =0$ and to ${\mathbb {G}}_m$ when $\lambda \in A^\times$, and let $\psi ^{(l)}:\mathcal {G}^{(\lambda )}\rightarrow \mathcal {G}^{(\lambda ^{p^l})}$ denote the $l$-th Kummer-Artin-Schreier type homomorphism determined by $\lambda$. We construct the Cartier dual of $\textrm{Ker}(\psi ^{(l)})$ over a $\mathbb {Z}/p^n\mathbb {Z}$-algebra.

   363-376
4. Existence of Optimal Solutions for Learning-based PDEs Governed by Time-Dependent $p$-Laplace and 1-Laplace Operators Van Thanh Nguyen , Thuan Quang Thai , Minh Hien Huynh PDF

   ▸ Abstract

   Learning-based partial differential equations, with applications in image processing and computer vision, are often formulated as optimal control problems constrained by PDEs. In this paper, we consider PDE constraints governed by the combination of the time-dependent p-Laplace and singular 1-Laplace type operators. Models of this type–and more general formulations–have been proposed and investigated numerically in Liu et al. [15, 16]. Theoretical results on the existence of optimal solutions have so far been established only for constant-coefficient operators in recent work [17]. The extension to time-dependent coefficients is more delicate, due to the lack of suitable stability results for nonlinear semigroups generated by the sum of two time-dependent monotone operators, together with the degeneracy of the multi-valued 1-Laplace operator. The main contribution of this paper is to address these issues and to establish the existence of optimal solutions for a class of singular PDE-constrained optimal control problems governed by the sum of time-dependent p-Laplace and multi-valued 1-Laplace operators.

   377-399
5. Limit Theorems for the Number of Up-crossings of the Conjunction Set of Stationary Gaussian Processes Viet-Hung Pham PDF

   ▸ Abstract

   In this paper, we investigate the limit theorem for the number of up-crossings of the conjunction set of stationary Gaussian processes. We prove that as the level tends to infinity then the normalized point process of the number of up-crossings converges weakly to a standard Poisson process. Our proof is based on a discretization argument and a Normal comparison lemma for the order statistics.

   401-413
6. Gradient Estimates for the Heat Kernel of Schrödinger Operators on Stratified Lie Groups and Their Applications Tran Phuoc An , Nguyen Ngoc Trong , Huynh Cao Truong PDF

   ▸ Abstract

   In this paper, we investigate the pointwise $L^\infty$ estimates and Hölder continuity of the difference between the gradients of the heat kernels of $-\Delta$ and the Schrödinger operators $-\Delta + \mathbb {V}$ on a Lie group, where $\mathbb {V}$ is a non-negative polynomial potential. These results have significant implications for gradient estimates of solutions to the parabolic Schrödinger equation. Consequently, we prove the boundedness of the Riesz transform on the $\textrm{BMO}$-type space associated with $\mathbb {L}$ and provide an application to the regularity of solutions of the Schrödinger equation. Some of these findings are novel even in the Euclidean setting $\mathbb {R}^n$.

   415-447
7. Stability Estimates for the Heisenberg Uncertainty Principle with Dunkl Weights Nguyen Tuan Duy , Nguyen Van Phong , Lam Thi Bich Thuy PDF

   ▸ Abstract

   We study the stability of the Heisenberg Uncertainty Principle with Dunkl weights, inspired by recent progress on stability in functional and geometric inequalities. First, using the Bakry-Émery $\varGamma _2$-calculus, we establish several versions of the Poincaré inequality for measures involving Dunkl-Gaussian weights. We then prove a Heisenberg-type uncertainty principle adapted to Dunkl weights through suitable identities. By combining these results, we obtain several stability estimates for the Dunkl-weighted Heisenberg uncertainty principle. Our work extends existing methods developed for monomial weights to the more complex Dunkl setting.

   449-462
8. Global and Nonglobal Solutions for a Mixed Local-nonlocal Heat Equation Brandon Carhuas-Torre , Ricardo Castillo , Ricardo Freire , Alex Lira , Miguel Loayza PDF

   ▸ Abstract

   In this work, we establish optimal criteria characterizing the global and nonglobal existence of solutions to a semilinear parabolic equation with a general source term $f$. The equation is governed by the mixed operator $$\mathcal {L} = -\Delta + (-\Delta )^s, \quad \text {with } 0< s < 1.$$ In our analysis, we study supersolutions and obtain estimates on the semigroup generated by $\mathcal {L}$. In particular, we recover Fujita’s critical exponent $1 + \tfrac{2s}{N}$, previously established by Biagi, Punzo, and Vecchi [2], and by Del Pezzo and Ferreira [18].

   463-476
9. Study on Special Features of New h-functions of Some Simply Connected Planar Regions Arunmaran Mahenthiram , Piriyalucksan Paramesvarampillai PDF

   ▸ Abstract

   The $h$-function $h(r)$, or the harmonic-measure distribution function, is defined as the probability that a Brownian particle released from a basepoint $z_0$ in a region ${\varOmega }$ first leaves the region ${\varOmega }$ within distance $r$ of $z_0$. In this paper, we compute the $h$-function formulas explicitly in terms of $r$ for some new simply connected planar regions by using the method of conformal maps. Especially, we investigate the behaviour of these $h$-functions at certain values of $r$ that correspond to specific points on the boundary of the respective regions ${\varOmega }$. In some cases, we explain the asymptotic behaviour as r decreases to the minimum distance between the basepoint $z_0$ and the boundary.

   477-497