We define a new ribbon group action on ribbon graphs that uses a semidirect product of a permutation group and the original ribbon group of Ellis-Monaghan and Moffatt to take (partial) twists and duals, or twuals, of ribbon graphs. A ribbon graph is a fixed point of this new ribbon group action if and only if it is isomorphic to one of its (partial) twuals. This extends the original ribbon group action, which only used the canonical identification of edges, to the more natural setting of self-twuality up to isomorphism. We then show that every ribbon graph has in its orbit an orientable embedded bouquet and prove that the (partial) twuality properties of these bouquets propagate through their orbits. Thus, we can determine (partial) twualities via these one vertex graphs, for which checking isomorphism reduces simply to checking dihedral group symmetries. Finally, we apply the new ribbon group action to generate all self-trial ribbon graphs on up to seven edges, in contrast with the few, large, very high-genus, self-trial regular maps found by Wilson, and by Jones and Poultin. We also show how the automorphism group of a ribbon graph yields self-dual, -petrial or –trial graphs in its orbit, and produce an infinite family of self-trial graphs that do not arise as covers or parallel connections of regular maps, thus answering a question of Jones and Poulton.

We construct two bounded functional calculi for sectorial operators on Banach spaces, which enhance the functional calculus for analytic Besov functions, by extending the class of functions, generalising and sharpening estimates and adapting the calculus to the angle of sectoriality. The calculi are based on appropriate reproducing formulas, they are compatible with standard functional calculi and they admit appropriate convergence lemmas and spectral mapping theorems. To achieve this, we develop the theory of associated function spaces in ways that are interesting and significant. As consequences of our calculi, we derive several well-known operator norm estimates and provide generalisations of some of them.

This article is about Lehn–Lehn–Sorger–van Straten eightfolds $Z$ and their anti-symplectic involution $\iota$. When $Z$ is birational to the Hilbert scheme of points on a K3 surface, we give an explicit formula for the action of $\iota$ on the Chow group of $0$-cycles of $Z$. The formula is in agreement with the Bloch–Beilinson conjectures and has some non-trivial consequences for the Chow ring of the quotient.

We study the structure and compute the stable rank of $C^{*}$-algebras of finite higher-rank graphs. We completely determine the stable rank of the $C^{*}$-algebra when the $k$-graph either contains no cycle with an entrance or is cofinal. We also determine exactly which finite, locally convex $k$-graphs yield unital stably finite $C^{*}$-algebras. We give several examples to illustrate our results.

We call a semigroup $S$ weakly right noetherian if every right ideal of $S$ is finitely generated; equivalently, $S$ satisfies the ascending chain condition on right ideals. We provide an equivalent formulation of the property of being weakly right noetherian in terms of principal right ideals, and we also characterize weakly right noetherian monoids in terms of their acts. We investigate the behaviour of the property of being weakly right noetherian under quotients, subsemigroups and various semigroup-theoretic constructions. In particular, we find necessary and sufficient conditions for the direct product of two semigroups to be weakly right noetherian. We characterize weakly right noetherian regular semigroups in terms of their idempotents. We also find necessary and sufficient conditions for a strong semilattice of completely simple semigroups to be weakly right noetherian. Finally, we prove that a commutative semigroup $S$ with finitely many archimedean components is weakly (right) noetherian if and only if $S/\mathcal {H}$ is finitely generated.

In this paper, we study the existence and concentration behaviour of multi-peak standing waves for a singularly perturbed Davey–Stewartson system, which arises in the theory of shallow water waves. For this purpose, we first give a sharp threshold of the existence of ground-state solutions to the related limiting problem. Next, combining the penalization method and the regularity theory of elliptic equations, we construct a family of positive solutions concentrating around any prescribed finite set of local minima, possibly degenerate, of the potential. A feature of this analysis is that we do not need any uniqueness or non-degeneracy conditions for the limiting equation. To the best of our knowledge, this paper is the first study dealing with the study of concentrating solutions for Davey–Stewartson systems. We emphasize that with respect to the classical Schrödinger equation, the presence of a singular integral operator in the Davey–Stewartson system forces the implementation of new ideas to obtain the existence of multi-peak solutions.

We consider Akbarov's holomorphic version of the non-commutative Pontryagin duality for a complex Lie group. We prove, under the assumption that $G$ is a Stein group with finitely many components, that (1) the topological Hopf algebra of holomorphic functions on $G$ is holomorphically reflexive if and only if $G$ is linear; (2) the dual cocommutative topological Hopf algebra of exponential analytic functional on $G$ is holomorphically reflexive. We give a counterexample, which shows that the first criterion cannot be extended to the case of infinitely many components. Nevertheless, we conjecture that, in general, the question can be solved in terms of the Banach-algebra linearity of $G$.

Let $G$ be a finite group with cyclic Sylow $p$-subgroups, and let $k$ be a field of characteristic $p$. Then $H^{*}(BG;k)$ and $H_*(\Omega BG{{}^{{}^{\wedge }}_p};k)$ are $A_\infty$ algebras whose structure we determine up to quasi-isomorphism.

Let $\operatorname {\mathrm {{\rm G}}}(n)$ be equal to either $\operatorname {\mathrm {{\rm PO}}}(n,1),\operatorname {\mathrm {{\rm PU}}}(n,1)$ or $\operatorname {\mathrm {\textrm {PSp}}}(n,1)$ and let $\Gamma \leq \operatorname {\mathrm {{\rm G}}}(n)$ be a uniform lattice. Denote by $\operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}}$ the hyperbolic space associated to $\operatorname {\mathrm {{\rm G}}}(n)$, where $\operatorname {\mathrm {{\rm K}}}$ is a division algebra over the reals of dimension d. Assume $d(n-1) \geq 2$.

In this article we generalise natural maps to measurable cocycles. Given a standard Borel probability $\Gamma $-space $(X,\mu _X)$, we assume that a measurable cocycle $\sigma :\Gamma \times X \rightarrow \operatorname {\mathrm {{\rm G}}}(m)$ admits an essentially unique boundary map $\phi :\partial _\infty \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \times X \rightarrow \partial _\infty \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$ whose slices $\phi _x:\operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \rightarrow \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$ are atomless for almost every $x \in X$. Then there exists a $\sigma $-equivariant measurable map $F: \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \times X \rightarrow \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$ whose slices $F_x:\operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}} \rightarrow \operatorname {\mathrm {\mathbb {H}^m_{{\rm K}}}}$ are differentiable for almost every $x \in X$ and such that $\operatorname {\mathrm {\textrm {Jac}}}_a F_x \leq 1$ for every $a \in \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}}$ and almost every $x \in X$. This allows us to define the natural volume $\operatorname {\mathrm {\textrm {NV}}}(\sigma )$ of the cocycle $\sigma $. This number satisfies the inequality $\operatorname {\mathrm {\textrm {NV}}}(\sigma ) \leq \operatorname {\mathrm {\textrm {Vol}}}(\Gamma \backslash \operatorname {\mathrm {\mathbb {H}^n_{{\rm K}}}})$. Additionally, the equality holds if and only if $\sigma $ is cohomologous to the cocycle induced by the standard lattice embedding $i:\Gamma \rightarrow \operatorname {\mathrm {{\rm G}}}(n) \leq \operatorname {\mathrm {{\rm G}}}(m)$, modulo possibly a compact subgroup of $\operatorname {\mathrm {{\rm G}}}(m)$ when $m>n$.

Given a continuous map $f:M \rightarrow N$ between compact hyperbolic manifolds, we also obtain an adaptation of the mapping degree theorem to this context.

For a prime number p and a free profinite group S on the basis X, let $S_{\left (n,p\right )}$, $n=1,2,\dotsc ,$ be the p-Zassenhaus filtration of S. For $p>n$, we give a word-combinatorial description of the cohomology group $H^2\left (S/S_{\left (n,p\right )},\mathbb {Z}/p\right )$ in terms of the shuffle algebra on X. We give a natural linear basis for this cohomology group, which is constructed by means of unitriangular representations arising from Lyndon words.

We prove existence results of two solutions of the problem

\[ \begin{cases} L(u)+u^{m-1}=\lambda u^{p-1} & \text{in}\ \Omega,\\ u>0 & \text{in}\ \Omega,\\ u=0 & \text{on}\ \partial \Omega, \end{cases} \]

$L(v)=-\textrm {div}(M(x)\nabla v)$

$p\in (2,2^{*}]$

$\lambda$

$m$

$m\to +\infty$

We define stable homotopy refinements of Khovanov’s arc algebras and tangle invariants.

We expand the critical point for site percolation on the d-dimensional hypercubic lattice in terms of inverse powers of 2d, and we obtain the first three terms rigorously. This is achieved using the lace expansion.

In this paper, we investigate the existence and nonexistence of results for a class of Hamiltonian-Choquard-type elliptic systems. We show the nonexistence of classical nontrivial solutions for the problem

\[ \begin{cases} -\Delta u + u= ( I_{\alpha} \ast |v|^{p} )v^{p-1} \text{ in } \mathbb{R}^{N},\\ -\Delta v + v= ( I_{\beta} \ast |u|^{q} )u^{q-1} \text{ in } \mathbb{R}^{N}, \\ u(x),v(x) \rightarrow 0 \text{ when } |x|\rightarrow \infty, \end{cases} \]

$(N+\alpha )/p + (N+\beta )/q \leq 2(N-2)$

$N\geq 3$

$(N+\alpha )/p + (N+\beta )/q \geq 2N$

$N=2$

$I_{\alpha }$

$I_{\beta }$

\[ \begin{cases} -\Delta u + u= (I_{\alpha} \ast |v|^{\frac{\alpha}{2}+1})|v|^{\frac{\alpha}{2}-1}v + g(v) \hbox{ in } \mathbb{R}^{2},\\ - \Delta v + v= (I_{\beta} \ast |u|^{\frac{\beta}{2}+1})|u|^{\frac{\beta}{2}-1}u + f(u), \hbox{ in } \mathbb{R}^{2},\\ u,v \in H^{1}(\mathbb{R}^{2}), \end{cases} \]

$\alpha ,\,\beta \in (0,\,2)$

$f,\,g$

We construct the complete set of orders of growth and define on it the generalized entropy of a dynamical system. With this object, we provide a framework wherein we can study the separation of orbits of a map beyond the scope of exponential growth. We show that this construction is particularly useful for studying families of dynamical systems with vanishing entropy. Moreover, we see that the space of orders of growth in which orbits are separated is wilder than expected. We achieve this with different types of examples.

Motivated by analogous questions in the setting of Steiner triple systems and Latin squares, Nenadov, Sudakov and Wagner [Completion and deficiency problems, Journal of Combinatorial Theory Series B, 2020] recently introduced the notion of graph deficiency. Given a global spanning property $\mathcal P$ and a graph $G$, the deficiency $\text{def}(G)$ of the graph $G$ with respect to the property $\mathcal P$ is the smallest non-negative integer t such that the join $G*K_t$ has property $\mathcal P$. In particular, Nenadov, Sudakov and Wagner raised the question of determining how many edges an n-vertex graph $G$ needs to ensure $G*K_t$ contains a $K_r$-factor (for any fixed $r\geq 3$). In this paper, we resolve their problem fully. We also give an analogous result that forces $G*K_t$ to contain any fixed bipartite $(n+t)$-vertex graph of bounded degree and small bandwidth.

We prove some qualitative results about the p-adic Jacquet–Langlands correspondence defined by Scholze, in the $\operatorname {\mathrm {GL}}_2(\mathbb{Q}_p )$ residually reducible case, using a vanishing theorem proved by Judith Ludwig. In particular, we show that in the cases under consideration, the global p-adic Jacquet–Langlands correspondence can also deal with automorphic forms with principal series representations at p in a nontrivial way, unlike its classical counterpart.

Kostochka and Thomason independently showed that any graph with average degree $\Omega(r\sqrt{\log r})$ contains a $K_r$ minor. In particular, any graph with chromatic number $\Omega(r\sqrt{\log r})$ contains a $K_r$ minor, a partial result towards Hadwiger’s famous conjecture. In this paper, we investigate analogues of these results in the directed setting. There are several ways to define a minor in a digraph. One natural way is as follows. A strong $\overrightarrow{K}_{\!\!r}$ minor is a digraph whose vertex set is partitioned into r parts such that each part induces a strongly connected subdigraph, and there is at least one edge in each direction between any two distinct parts. We investigate bounds on the dichromatic number and minimum out-degree of a digraph that force the existence of strong $\overrightarrow{K}_{\!\!r}$ minors as subdigraphs. In particular, we show that any tournament with dichromatic number at least 2r contains a strong $\overrightarrow{K}_{\!\!r}$ minor, and any tournament with minimum out-degree $\Omega(r\sqrt{\log r})$ also contains a strong $\overrightarrow{K}_{\!\!r}$ minor. The latter result is tight up to the implied constant and may be viewed as a strong-minor analogue to the classical result of Kostochka and Thomason. Lastly, we show that there is no function $f\;:\;\mathbb{N} \rightarrow \mathbb{N}$ such that any digraph with minimum out-degree at least f(r) contains a strong $\overrightarrow{K}_{\!\!r}$ minor, but such a function exists when considering dichromatic number.

In this short article, we will be principally investigating two classes of modules over any given group ring – the class of Gorenstein projectives and the class of Benson's cofibrants. We begin by studying various properties of these two classes and studying some of these properties comparatively against each other. There is a conjecture made by Fotini Dembegioti and Olympia Talelli that these two classes should coincide over the integral group ring for any group. We make this conjecture over group rings over commutative rings of finite global dimension and prove it for some classes of groups while also proving other related results involving the two classes of modules mentioned.