The graphical operation of insplitting is key to understanding conjugacy of shifts of finite type (SFTs) in both one and two dimensions. In this paper, we consider two approaches to studying two-dimensional (2D) SFTs: textile systems and rank-2 graphs. Nasu’s textile systems describe all two-sided 2D SFTs up to conjugacy [Johnson and Madden, ‘The decomposition theorem for two-dimensional shifts of finite type’, Proc. Amer. Math. Soc. 127(5) (1999), 1533–1543; Aso, ‘Conjugacy of $Z^2$-subshifts and textile systems’, Publ. RIMS. Kyoto Univ 36 (2000), 1–18], whereas the 2-graphs (higher-rank graphs of rank 2) introduced by Kumjian and Pask yield associated $C^*$-algebras [Tang, ‘Tiling systems and 2-graphs associated to textile systems’, PhD Thesis, University of Wollongong, 2013, http://ro.uow.edu.au/theses/4003]. Both models have a naturally associated notion of insplitting (introduced for textile systems by Johnson and Madden [‘The decomposition theorem for two-dimensional shifts of finite type’, Proc. Amer. Math. Soc. 127(5) (1999), 1533–1543] and for 2-graphs by Eckhardt et al. [‘Moves on k-graphs preserving Morita equivalence’, Canad. J. Math. 74 (2022), 655–685]). We show that these notions do not coincide, raising the question of whether insplitting a 2-graph induces a conjugacy of the associated one-sided 2D SFTs. Our first main result shows how to reconstruct 2-graph insplitting using textile-system insplits and inversions, and consequently proves that 2-graph insplitting induces a conjugacy of dynamical systems. We also present several other facets of the relationship between 2-graph insplitting and textile-system insplitting. Incorporating an insplit of the ‘bottom’ graph of the textile system turns out to be key to this relationship. By articulating the connection between operator-algebraic and dynamical notions of insplitting in two dimensions, this article lays the groundwork for a $C^*$-algebraic framework for classifying one-sided conjugacy in higher-dimensional SFTs.

Given a permutation group G, the derangement graph of G is defined with vertex set G, where two elements x and y are adjacent if and only if $xy^{-1}$ is a derangement. We establish that if G is transitive with degree exceeding 30, then the derangement graph of G contains a complete subgraph with four vertices. In the process, we determine all transitive groups whose derangement graph does not contain a complete subgraph on four vertices. As a consequence, if G is a normal subgroup of A such that $|A : G| = 3$ and U is a subgroup of G satisfying $G = \bigcup _{a \in A} U^a$, then $|G : U| \leq 10$. This provides support for a conjecture by Neumann and Praeger concerning Kronecker classes.

In this paper, we show the existence, uniqueness and stability of nontrivial solutions to the following Minkowski-curvature problems on unbounded domains:

$$ \begin{align*} \bigg(\frac{x'}{\sqrt{1-x^{\prime2}}}\bigg)'=f(t,\ x),\quad t\geq t_0, \quad \lim_{t\to\infty}x(t)=\psi_0,\quad \lim_{t\to\infty}x'(t)e^{t}=0, \end{align*} $$

$f:\ [t_0, \infty )\times \mathbb {R}\rightarrow \mathbb {R}$

$t_0>0$

$\psi _0\in \mathbb {R}$

We study the problem of improving Dirichlet’s theorem of metric Diophantine approximation in the most general and multiplicative form in the $S$-adic setting. Our approach is based on translation of the problem related to Dirichlet improvability into a dynamical one, and the main technique of our proof is the $S$-adic version of quantitative nondivergence estimate due to Kleinbock and Tomanov. The main result of this paper can be regarded as the number field version of earlier works of Kleinbock and Weiss [‘Dirichlet’s theorem on Diophantine approximation and homogeneous flows’, J. Mod. Dyn. 4 (2008), 43–62], and of the second named author and Ghosh [‘Dirichlet’s theorem in function fields’, Canad. J. Math. 69(3) (2017), 532–547]. Also, this in turn generalizes a result of Datta and Radhika [‘Singular Vectors on Manifolds over Number Fields’, Monatsh. Math. 200(3) (2023), 545–568] on singularity of vectors to any number field $K$ and $S$ containing all Archimedian places.

In this paper, we develop a continuity method for the Dirichlet problem of Hessian equations on Riemannian manifolds. Such equations, introduced by Caffarelli et al. [‘The Dirichlet problem for nonlinear second-order elliptic equations III: functions of eigenvalues of the Hessians’, Acta Math. 155 (1985), 261–301] are defined in terms of the eigenvalues of the Hessian and a given pair $(f,\Gamma )$, where f is a symmetric function defined in a symmetric cone $\Gamma \subset \mathbb {R}^n$ and $\Gamma $ specifies the set of admissible eigenvalues for the solution. Our method combines techniques from Morse theory with a characterization of the pair $(f,\Gamma )$. More precisely, in the type-2 case, we first construct admissible functions using Morse theory and then solve the Dirichlet problem without any additional assumptions on the boundary or the subsolution. Building on this characterization of the pair, we can approximate the type-1 equation by a family of type-2 equations.