The averaged distance structure of one-dimensional regular model sets is determined via their pair-correlation functions. The latter lead to covariograms and cross covariograms of the windows, which give continuous functions in internal space. While they are simple tent-shaped, piecewise linear functions for intervals, the typical case for inflation systems leads to convolutions of Rauzy fractals, which are difficult to compute. In the presence of an inflation structure, an alternative path is possible via the exact renormalisation structures of the pair-correlation functions. We introduce this approach and derive two concrete examples, which display unexpectedly complex and wild behaviour.

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.

Let $\{a_n(x)\}_{n\geq 1}$ be the sequence of digits of $x\in (0,1)$ in an infinite iterated function system with polynomial decay of the derivative. We first study the multifractal spectrum of the convergence exponent, defined by the sequence of digits $\{a_n(x)\}_{n\geq 1}$ and the weighted products of distinct digits with finite numbers, and then calculate the Hausdorff dimensions of the intersections of sets defined by the convergence exponent of the weighted product of distinct digits with finite numbers and sets of points whose digits are nondecreasing in such iterated function systems.

In a previous paper, we stated and motivated counting conjectures for fusion systems that are purely local analogues of several local-to-global conjectures in the modular representation theory of finite groups. Here, we verify some of these conjectures for fusion systems on an extraspecial group of order $p^3$, which contain among them the Ruiz–Viruel exotic fusion systems at the prime $7$. As a byproduct, we verify Robinson’s ordinary weight conjecture for principal p-blocks of almost simple groups G realizing such (nonconstrained) fusion systems.

We study the well-posedness of solutions to the general nonlinear parabolic equations with merely integrable data in time-dependent Musielak–Orlicz spaces. With the help of a density argument, we establish the existence and uniqueness of both renormalized and entropy solutions. Moreover, we conclude that the entropy and renormalized solutions for this equation are equivalent. Our results cover a variety of problems, including those with Orlicz growth, variable exponents and double-phase growth.

The primary objective of this paper is to show how the theory of groupoids and their $C^*$-algebras provide new proofs and extensions of Mallat’s famous theorems on the construction of multiresolution analyses that form the basis of wavelet theory. This work was inspired in large part by the research of Iain Raeburn and co-authors.

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 $\mathcal {P}$.