Gcd-graphs over the ring of integers modulo n are a simple and elegant class of integral graphs. The study of these graphs connects multiple areas of mathematics, including graph theory, number theory, and ring theory. In a recent work (J. Mináč, T. T. Nguyen and N. D. Tân, ‘On the gcd-graphs over polynomial rings’, Canad. J. Math., to appear. Published online (1 October 2025)), inspired by the analogy between number fields and function fields, we define and study gcd-graphs over polynomial rings with coefficients in finite fields. We discover that, in both cases, gcd-graphs share many similar and analogous properties. In this article, we extend this line of research further. Among other topics, we explore an analog of a conjecture of So and a weaker version of Sander and Sander concerning the conditions under which two gcd-graphs are isomorphic or isospectral. We also provide several constructions showing that, unlike the case over $\mathbb {Z}$, it is not uncommon for two gcd-graphs over polynomial rings to be isomorphic.

In this paper, we study affine reflection subgroups and their root subsystems in arbitrary infinite Coxeter groups of finite rank. We give a characterization of limit roots (the accumulation points of the projections of roots onto suitably chosen hyperplanes) associated with affine reflection subgroups. We describe the relationship between these limit roots and all other limit roots. In particular, we show that each limit root associated with an affine reflection subgroup may only produce positive values with finitely many positive roots under the bilinear form afforded by the Tits representation (thus satisfying a so-called Pos-finite condition); and we show that an arbitrary limit root producing positive bilinear form values with only finitely many positive roots must be a convex combination of limit roots associated with affine reflection subgroups. As an application of this Pos-finite condition, we give an independent characterization of when a finitely generated infinite Coxeter group may possess affine reflection subgroups. We also use this Pos-finite analysis to partially resolve the question of whether the projections of an infinite injective sequence of roots totally ordered by the dominance partial ordering possesses or not a unique point of accumulation.

This paper deals with the mean first escape time of Brownian motion on asymptotically hyperbolic and gas giant surfaces. We show that for a boundary defining function $\rho $, the mean first escape time $u_\varepsilon (x)$ from the truncated Riemannian surface with an asymptotically hyperbolic metric $(M_\varepsilon ,\bar {g}/\rho ^2) = (\{x\in M:\rho (x)> \varepsilon \},\bar {g}/\rho ^2) \subset (M,\bar {g}/\rho ^2)$ satisfies the asymptotic expansion $u_\varepsilon (x) = -\log \varepsilon + \mathcal {O}(1)$ as $\varepsilon \to 0 $. Furthermore, we show that in the case of a gas giant metric $g = \bar {g}/\rho ^\alpha $, where $\alpha \in (0,2)$, the mean first escape time from the surface $(M_\varepsilon ,\bar {g}/\rho ^\alpha )$ satisfies $u_\varepsilon (x) = \mathcal {O}(1)$ as $\varepsilon \to 0 $. Using techniques from the theory of polyhomogeneous conormal functions, we explain this difference between the mean first escape time on gas giant metric surfaces and asymptotically hyperbolic surfaces on the unit disc.

This paper presents various transcendence results in the ring of integers modulo infinitely large primes $\mathcal {A}$. In the ring $\mathcal {A}$, one can consider two notions of transcendence. One is based on the notion of finite algebraic numbers introduced by Rosen, while the other is transcendence in the naive sense. It is known that transcendence in the latter sense automatically implies transcendence in the former sense. In this paper, we strengthen results due to Anzawa and Funakura and to Luca and Zudilin by removing some of their assumptions and, in some cases, upgrading them to statements of naive transcendence. We also present several examples of naive transcendental numbers that do not seem to have appeared previously in the literature. Although we are not able to establish naive transcendence for certain numbers, we prove the irrationality of numbers such as $\log _{\mathcal {A}}(2)$ under the ABC conjecture.

We completely characterize insoluble groups whose subgroups are pronormal or pronormalize every subgroup. We also completely describe $p$-groups with this property and prove many structural results in the soluble case (for example, such groups are always of length at most $3$ and have Sylow towers).

Reciprocality in Kirchberg algebras with finitely generated $\mathrm {K}$-groups is regarded as a $\mathrm {K}$-theoretic duality through $\mathrm {K}$-groups and strong extension groups. We prove that the reciprocal Kirchberg algebra has a universal property with respect to some generating $C^*$-subalgebra and a family of generating partial isometries. By using the universal property, we prove that there exists an aperiodic ergodic automorphism on an arbitrary unital Kirchberg algebra with finitely generated $\mathrm {K}$-groups, which has a unique invariant state. The state is pure.

We give a length-one projective resolution of the trivial module for the groupoid of a semi-saturated partial action (in the sense of Exel) of a free group on a compact Hausdorff and totally disconnected space. As a consequence, we obtain an elementary computation of the homology of these groupoids, which include transformation groupoids of free group actions and Deaconu–Renault groupoids of systems $(X,T)$, where $X$ is compact Hausdorff and totally disconnected, and $T$ is a local homeomorphism with domain a clopen subset of $X$. We also show that algebra of such a partial action groupoid over a field has global dimension at most $2$ when the space is second countable.

We show that the $C^*$-algebra of a countable directed graph is singly generated. As a consequence, any $C^*$-algebra generated by a countable family of projections and partial isometries satisfying Cuntz–Krieger relations is singly generated.

Our aim in this article is to establish new characterizations of hyperplanes in Euclidean space via mild constraints on the higher-order anisotropic mean curvatures of an immersed complete hypersurface. Our approach is based on the study of the behavior of the height and angle functions naturally attached to a hypersurface, jointly with the use of appropriate maximum principles dealing with the notions of convergence at infinity and polynomial volume growth. In this context, we also obtain lower estimates for the minimum anisotropic relative null index of an immersed complete hypersurface, which can be regarded as a measure of how close such a hypersurface is to being flat.

In this paper, we establish a lower bound for the maxima of derivatives of the Dedekind zeta function of a cyclotomic field on the critical line. Employing a double-version convolution formula and combined with special GCD sums, our result generalizes the work of Bondarenko et al. [‘A dichotomy for extreme values of zeta and Dirichlet L-functions’, Bull. Lond. Math. Soc. 55 (2023), 2963–2975]. We also establish a lower bound via the resonance method when the real part is near the critical line; both of the above results refine parts of Yang’s [‘Extreme values of derivatives of the Riemann zeta function’, Mathematika 68 (2022), 486–510] work.

A finitely based, finitely generated variety with finitely many subvarieties is a Cross variety. In the present article, it is shown that a variety of J-trivial monoids is Cross if and only if it excludes 14 specific almost Cross varieties. Consequently, these 14 varieties exhaust all almost Cross varieties of J-trivial monoids.

In this paper, we define and study noncommutative affine pencils of conics, and give a complete classification result. We also fully classify four-dimensional Frobenius algebras. It turns out that the classification of noncommutative affine pencils of conics is the same as the classification of four-dimensional Frobenius algebras.

Given any rectangular polyhedron $3$-manifold $\mathcal {P}$ tiled with unit cubes, we find infinitely many explicit directions related to cubic algebraic numbers such that all half-infinite geodesics in these directions are uniformly distributed in $\mathcal {P}$.