Geodesic flows and the mother of all continued fractions

We extend the Series [The modular surface and continued fractions, J. London Math. Soc. (2) 31(1) (1985) 69–80] connection between the modular surface [Formula: see text], cutting sequences, and regular continued fractions to the slow converging Lehner and Farey continued fractions with digits [Formula: see text] and [Formula: see text] in the notation used for the Lehner continued fractions. We also introduce an alternative insertion and singularization algorithm for Farey expansions and other non-semiregular continued fractions, and an alternative dual expansion to the Farey expansions so that [Formula: see text] is invariant under the natural extension map.

Isomorphisms of subcategories of fusion systems of blocks and Clifford theory

Let k be an algebraically closed field of prime characteristic p. Let G be a finite group, let N be a normal subgroup of G, and let c be a G-stable block of kN so that {(kN)c} is a p-permutation G-algebra. As in Section 8.6 of [M. Linckelmann, The Block Theory of finite Group Algebras: Volume 2, London Math. Soc. Stud. Texts 92, Cambridge University, Cambridge, 2018], a {(G,N,c)}-Brauer pair {(R,f_{R})} consists of a p-subgroup R of G and a block {f_{R}} of {(kC_{N}(R))}. If Q is a defect group of c and {f_{Q}\in\operatorname{\textit{B}\ell}(kC_{N}(Q))}, then {(Q,f_{Q})} is a {(G,N,c)}-Brauer pair. The {(G,N,c)}-Brauer pairs form a (finite) poset. Set {H=N_{G}(Q,f_{Q})} so that {(Q,f_{Q})} is an {(H,C_{N}(Q),f_{Q})}-Brauer pair. We extend Lemma 8.6.4 of the above book to show that if {(U,f_{U})} is a maximal {(G,N,c)}-Brauer pair containing {(Q,f_{Q})}, then {(U,f_{U})} is a maximal {(H,C_{N}(c),f_{Q})}-Brauer pair containing {(Q,f_{Q})} and conversely. Our main result shows that the subcategories of {\mathcal{F}_{(U,f_{U})}(G,N,c)} and {\mathcal{F}_{(U,f_{U})}(H,C_{N}(Q),f_{Q})} of objects between and including {(Q,f_{Q})} and {(U,f_{U})} are isomorphic. We close with an application to the Clifford theory of blocks.

Completeness in Quasi-Pseudometric Spaces—A Survey

The aim of this paper is to discuss the relations between various notions of sequential completeness and the corresponding notions of completeness by nets or by filters in the setting of quasi-metric spaces. We propose a new definition of right K-Cauchy net in a quasi-metric space for which the corresponding completeness is equivalent to the sequential completeness. In this way we complete some results of R. A. Stoltenberg, Proc. London Math. Soc. 17 (1967), 226–240, and V. Gregori and J. Ferrer, Proc. Lond. Math. Soc., III Ser., 49 (1984), 36. A discussion on nets defined over ordered or pre-ordered directed sets is also included.

Universally Koszul and initially Koszul properties of Orlik–Solomon algebras

Let [Formula: see text] be a field with [Formula: see text] and [Formula: see text] an exterior algebra over [Formula: see text] with a standard grading [Formula: see text]. Let [Formula: see text] be a graded algebra, where [Formula: see text] is a graded ideal in [Formula: see text]. In this paper, we study universally Koszul and initially Koszul properties of [Formula: see text] and find classes of ideals [Formula: see text] which characterize such properties of [Formula: see text]. As applications, we classify arrangements whose Orlik–Solomon algebras are universally Koszul or initially Koszul. These results are related to a long-standing question of Shelton–Yuzvinsky [B. Shelton and S. Yuzvinsky, Koszul algebras from graphs and hyperplane arrangements, J. London Math. Soc. 56 (1997) 477–490].

DIOPHANTINE APPROXIMATION WITH GAUSSIAN PRIMES

In this paper we prove that the exact analogue of the author’s work with real irrationals and rational primes (G. Harman, On the distribution of $\alpha p$ modulo one II, Proc. London Math. Soc. (3) 72, 1996, 241–260) holds for approximating $\alpha \in \mathbb{C}\setminus \mathbb{Q}[i]$ with Gaussian primes. To be precise, we show that for such $\alpha $ and arbitrary complex $\beta $ there are infinitely many solutions in Gaussian primes $p$ to $$\begin{equation*} ||\alpha p + \beta|| <| p|^{-7/22}, \end{equation*}$$where $||\cdot ||$ denotes distance to a nearest member of $\mathbb{Z}[i]$. We shall, in fact, prove a slightly more general result with the Gaussian primes in sectors, and along the way improve a recent result due to Baier (S. Baier, Diophantine approximation on lines in $\mathbb{C}^2$ with Gaussian prime constraints, Eur. J. Math. 3, 2017, 614–649).

The global dimension of the algebras of polynomial integro-differential operators 𝕀n and the Jacobian algebras 𝔸n

The aim of the paper is to prove two conjectures from the paper [V. V. Bavula, The algebra of integro-differential operators on a polynomial algebra, J. London Math. Soc. (2) 83 (2011) 517–543, arXiv:math.RA/0912.0723] that the (left and right) global dimension of the algebra [Formula: see text] of polynomial integro-differential operators and the Jacobian algebra [Formula: see text] is equal to [Formula: see text] (over a field of characteristic zero). The algebras [Formula: see text] and [Formula: see text] are neither left nor right Noetherian and [Formula: see text]. Furthermore, they contain infinite direct sums of nonzero left/right ideals and are not domains. An analogue of Hilbert’s Syzygy Theorem is proven for the algebras [Formula: see text], [Formula: see text] and their factor algebras. It is proven that the global dimension of all prime factor algebras of the algebras [Formula: see text] and [Formula: see text] is [Formula: see text] and the weak global dimension of all the factor algebras of [Formula: see text] and [Formula: see text] is [Formula: see text].

Local triple derivations from C$^*$-algebras into their iterated duals

We show that every local triple derivation from a C$^*$-algebra into any of its iterated duals is a triple derivation. This result partially solves a problem posed by M. Burgos \emph{et al.} in [Bull. London Math. Soc. 46 (4), 709-724 (2014)].

Coloring 3-power of 3-subdivision of subcubic graph

Let [Formula: see text] be a graph and [Formula: see text] be a positive integer. The [Formula: see text]-subdivision [Formula: see text] of [Formula: see text] is the graph obtained from [Formula: see text] by replacing each edge by a path of length [Formula: see text]. The [Formula: see text]-power [Formula: see text] of [Formula: see text] is the graph with vertex set [Formula: see text] in which two vertices [Formula: see text] and [Formula: see text] are adjacent if and only if the distance [Formula: see text] between [Formula: see text] and [Formula: see text] in [Formula: see text] is at most [Formula: see text]. Note that [Formula: see text] is the total graph [Formula: see text] of [Formula: see text]. The chromatic number [Formula: see text] of [Formula: see text] is the minimum integer [Formula: see text] for which [Formula: see text] has a proper [Formula: see text]-coloring. The total chromatic number of [Formula: see text], denoted by [Formula: see text], is the chromatic number of [Formula: see text]. Rosenfeld [On the total coloring of certain graphs, Israel J. Math. 9 (1971) 396–402] and independently, Vijayaditya [On total chromatic number of a graph, J. London Math. Soc. 2 (1971) 405–408] showed that for a subcubic graph [Formula: see text], [Formula: see text]. In this note, we prove that for a subcubic graph [Formula: see text], [Formula: see text].

The mapping class group orbits in the framings of compact surfaces

We compute the mapping class group orbits in the homotopy set of framings of a compact connected oriented surface with non-empty boundary. In the case g≥2, the computation is some modification of Johnson’s results (D. Johnson, Spin structures and quadratic forms on surfaces, J. London Math. Soc. (2)22 (1980), 365–373; D. Johnson, An abelian quotient of the mapping class group ℐg, Math. Ann.249 (1980), 225–242) and certain arguments on the Arf invariant, while we need an extra invariant for the genus 1 case. In addition, we discuss how this invariant behaves in the relative case, which Randal-Williams (O. Randal-Williams, Homology of the moduli spaces and mapping class groups of framed, r-Spin and Pin surfaces, J. Topology7 (2014), 155–186) studied for g≥2.