We consider a subshift of finite type endowed with a Markov measure that is given by a stochastic matrix. We introduce a Markov hole determined by a finite collection of allowed words in the subshift. We first present a simple yet precise formula to compute the escape rate into the hole as the spectral radius of a perturbed stochastic matrix, where the rule of perturbation is governed by the hole. The combinatorial nature of the subshift comes to our aid in obtaining another formulation of the escape rate as the logarithm of the smallest real pole of a certain rational function, by way of recurrence relations. This proves crucial in comparing the escape rates into cylinders based at words of fixed length. Merits of both the formulas are illustrated through examples.

The notion of weighted $\alpha $-composition was introduced by Ruhan Zhao in the 1990s. In this paper, we study several analytic function spaces that are closely related to weighted $\alpha $-composition. These include $\alpha $-Bloch spaces, $F(p,q,s)$ spaces, and Campanato spaces. We obtain derivative-free characterizations for $\alpha $-Bloch spaces and $F(p,q,s)$ spaces, which improve some previous results in the literature. We also obtain a certain version of Carleson measures for Campanato spaces and $F(p,q,s)$ spaces.

This paper proves that the integrality of algebraic Witt vectors over imaginary quadratic fields is decidable; based on this, some related problems are also discussed.

We introduce the notion of a Nakajima bundle representation. Given a labelled quiver and a variety or manifold X, such a representation involves an assignment of a complex vector bundle on X to each node of the double quiver; to the edges, we assign sections of, and connections on, associated twisted bundles. We for the most part restrict attention in our development to algebraic curves or Riemann surfaces. Our construction simultaneously generalizes ordinary Nakajima quiver representations on the one hand and quiver bundles on the other hand. These representations admit gauge-theoretic characterizations, analogous to the Atiyah–Drinfel’d–Hitchin–Manin equations in the original work of Nakajima, allowing for the construction of these generalized quiver varieties using a reduction procedure with moment maps. We study the deformation theory of Nakajima bundle representations, prove a Hitchin–Kobayashi correspondence between such representations and stable quiver bundles, examine the natural torus action on the resulting moduli varieties, and comment on scenarios where the variety is hyperkähler. Finally, we produce concrete examples that recover known moduli spaces.

In this paper, we describe étale Boolean right restriction monoids in terms of Boolean inverse monoids. We show that the Thompson groups arise naturally in this context.