Commutator blueprints can be seen as blueprints for constructing RGD systems over $\mathbb {F}_2$ with prescribed commutation relations. In this paper, we construct several families of Weyl-invariant commutator blueprints, mostly of universal type. Also applying another result of the author, we obtain new examples of exotic RGD systems of universal type over $\mathbb {F}_2$. In particular, we generalize Tits’ construction of uncountably many trivalent Moufang twin trees to higher rank, we obtain an example of an RGD system of rank $3$ such that the nilpotency degree of the groups $U_w$ is unbounded, and we construct a commutator blueprint of type $(4, 4, 4)$ that is used to answer a question of Tits from the late $1980$s about twin buildings.

Levels of cancellativity in commutative monoids M, determined by stable-rank values in $\mathbb {Z}_{> 0} \cup \{\infty \}$ for elements of M, are investigated. The behavior of the stable ranks of multiples $ka$, for $k \in \mathbb {Z}_{> 0}$ and $a \in M$, is determined. In the case of a refinement monoid M, the possible stable-rank values in archimedean components of M are pinned down. Finally, stable rank in monoids built from isomorphism or other equivalence classes of modules over a ring is discussed.

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.

Let A be an abelian variety defined over a global function field F and let p be a prime distinct from the characteristic of F. Let $F_\infty $ be a p-adic Lie extension of F that contains the cyclotomic $\mathbb {Z}_p$-extension $F^{\mathrm {cyc}}$ of F. In this paper, we investigate the structure of the p-primary Selmer group $\mathrm {Sel}(A/F_\infty )$ of A over $F_\infty $. We prove the $\mathfrak {M}_H(G)$-conjecture for $A/F_\infty $. Furthermore, we show that both the $\mu $-invariant of the Pontryagin dual of the Selmer group $\mathrm {Sel}(A/F^{\mathrm {cyc}})$ and the generalized $\mu $-invariant of the Pontryagin dual of the Selmer group $\mathrm {Sel}(A/F_\infty )$ are zero, thereby proving Mazur’s conjecture for $A/F$. We then relate the order of vanishing of the characteristic elements, evaluated at Artin representations, to the corank of the Selmer group of the corresponding twist of A over the base field F. Assuming the finiteness of the Tate–Shafarevich group, we establish that this corank equals the order of vanishing of the L-function of $A/F$ at $s=1$. Finally, we extend a theorem of Sechi—originally proved for elliptic curves without complex multiplication—to abelian varieties over global function fields. This is achieved by adapting the notion of generalized Euler characteristic, introduced by Zerbes for elliptic curves over number fields. This new invariant allows us, via Akashi series, to relate the generalized Euler characteristic of $\mathrm {Sel}(A/F_\infty )$ to the Euler characteristic of $\mathrm {Sel}(A/F^{\mathrm {cyc}})$.

We prove that Brinkmann’s problems are decidable for endomorphisms of $F_n\times F_m$: given $(x,y),(z,w)\in F_n\times F_m$ and $\Phi \in \mathrm {End}(F_n\times F_m)$, it is decidable whether there is some $k\in \mathbb {N}$ such that $(x,y)\Phi ^k=(z,w)$ (or $(x,y)\Phi ^k\sim (z,w)$). We also prove decidability of a two-sided version of Brinkmann’s conjugacy problem for injective endomorphisms which, from the work of Logan, yields a solution to the conjugacy problem in ascending HNN-extensions of $F_n\times F_m$. Finally, we study the dynamics of automorphisms of $F_n\times F_m$ at the infinity, proving that that their dynamics at the infinity is asymptotically periodic, as occurs in the free and free-abelian times free cases.

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.

A ${\mathcal Z}$-subalgebra $U_{\mathcal Z}^\jmath {(n)}$ (${\mathcal Z}=\mathbb Z[\upsilon ,\upsilon ^{-1}]$) for the i-quantum group ${\mathbf {U}}^{\jmath }(n)$ over the field $\mathbb Q(\upsilon )$ is constructed by two of the authors [‘A new realisation of the i-quantum group $U^\jmath {(n)}$’, J. Pure Appl. Algebra 226(1) (2022), Paper no. 106793, 27 pages, Theorem 6.5], using a Beilinson–Lusztig–MacPherson (BLM) type realisation. In this paper, we construct bases for $U_{\mathcal Z}^\jmath {(n)}$, including the monomial basis conjectured in [‘A new realisation of the i-quantum group $U^\jmath {(n)}$’, J. Pure Appl. Algebra 226(1) (2022), Paper no. 106793, 27 pages, Remark 6.6(4)]. This proves that the ${\mathcal Z}$-algebra $U_{\mathcal Z}^\jmath {(n)}$ is a free ${\mathcal Z}$-module. Hence, $U_{\mathcal Z}^\jmath {(n)}$ is in fact an integral form of Lusztig type. This construction is further extended to the i-quantum hyperalgebra over a field of any characteristic. By specialising $\upsilon $ to an l th primitive root $\varepsilon $ of $1$ with l odd, a realisation of the quotient of modulo the ideal generated by $d_i^l-1$, for all $1\leqslant i\leqslant n+1$, is also given as a by-product.

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.

Special Ricci–Hessian equations on Kähler manifolds $(M,g)$, as defined by Maschler [‘Special Kähler–Ricci potentials and Ricci solitons’, Ann. Global Anal. Geom. 34 (2008), 367–380], involve functions $\tau $ on M and state that, for some function $\alpha $ of the real variable $\tau\kern-0.8pt $, the sum of $\alpha \nabla d\tau\kern-0.8pt $ and the Ricci tensor equals a functional multiple of the metric g, while $\alpha \nabla d\tau\kern-0.8pt $ itself is assumed to be nonzero almost everywhere. Three well-known obvious ‘standard’ cases are provided by (non-Einstein) gradient Kähler–Ricci solitons, conformally-Einstein Kähler metrics, and special Kähler–Ricci potentials. We show that, outside of these three cases, such an equation can only occur in complex dimension two and, at generic points, it must then represent one of three types, for which, up to normalizations, $\alpha =2\cot \tau\kern-0.8pt $, $\alpha =2\coth \tau\kern-0.8pt $, or $\alpha =2\tanh \tau\kern-0.8pt $. We also use the Cartan–Kähler theorem to prove that these three types are actually realized in a ‘nonstandard’ way.

We want to compute generic $\mathrm {Ext}$-spaces of twisted polynomial functors in relation to the $\mathrm {Ext}$-spaces of the untwisted ones. We are able to conjecture a direct formula that is suggestively similar to the one known for ungraded functors. Thanks to the study of a spectral sequence, we get to a total computation in low degrees, with remarkable consequences at the level of generic cohomology. Moreover, a partial computation in all degrees proves the inclusion of a big direct summand and the existence of a suggestive universal class.

We introduce a generating function approach to the affine Brauer and Kauffman categories, and show how it allows one to efficiently recover important sets of relations in these categories. We use this formalism to deduce restrictions on possible categorical actions and show how this recovers admissibility results that have appeared in the literature on cyclotomic Birman–Murakami–Wenzl (BMW) algebras and their degenerate versions, also known as cyclotomic Nazarov–Wenzl algebras or VW algebras.

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.

Let $\{u_n\}_n$ be a nondegenerate linear recurrence sequence of integers with Binet’s formula given by ${u_n= \sum _{i=1}^{m} P_i(n)\alpha _i^n.}$ Assume $\max _i \vert \alpha _i \vert>1$. In 1977, Loxton and Van der Poorten conjectured that for any $\epsilon>0$, there is an effectively computable constant $C(\epsilon )$ such that if $ \vert u_n \vert < (\max _i\{ \vert \alpha _i \vert \})^{n(1-\epsilon )}$, then $n<C(\epsilon )$. Using results of Schmidt and Evertse, a complete noneffective (qualitative) proof of this conjecture was given by Fuchs and Heintze [‘On the growth of linear recurrences in function fields’, Bull. Aust. Math. Soc. 104(1) (2021), 11–20] and, independently, by Karimov et al. [‘The power of positivity’, Proc. LICS 23 (2023), 1–11]. In this paper, we give an effective upper bound for the number of solutions of the inequality $\vert u_n \vert < (\max _i\{ \vert \alpha _i \vert \})^{n(1-\epsilon )}$, thus extending several earlier results by Schmidt, Schlickewei and Van der Poorten.

Let ${\mathrm {U}}_n({\mathbb {F}}_q)$ be the unitriangular group and ${\mathrm {U}}_{a,b,c,d}({\mathbb {F}}_q)$ the four-block unipotent radical of the standard parabolic subgroup of $\mathrm {GL}_{n}$, where $a+b+c+d=n$. In this paper, we study the class of all pattern subgroups of ${\mathrm {U}}_{a,b,c,d}({\mathbb {F}}_{q})$. We establish character-number formulae of degree $q^e$ for all these pattern groups. For pattern subgroups $G_{{\mathcal {D}}_m}({\mathbb {F}}_q)$ in this class, we provide an algebraic geometric approach to their polynomial properties, which verifies an analogue of Lehrer’s conjecture for these pattern groups.

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.