Let $\Sigma _{g,p}$ be the genus–g oriented surface with p punctures, with either g > 0 or p > 3. We show that $MCG(\Sigma _{g,p})/DT$ is acylindrically hyperbolic where DT is the normal subgroup of the mapping class group $MCG(\Sigma _{g,p})$ generated by $K^{th}$ powers of Dehn twists about curves in $\Sigma _{g,p}$ for suitable K.

Moreover, we show that in low complexity $MCG(\Sigma _{g,p})/DT$ is in fact hyperbolic. In particular, for 3g − 3 + p ⩽ 2, we show that the mapping class group $MCG(\Sigma _{g,p})$ is fully residually non-elementary hyperbolic and admits an affine isometric action with unbounded orbits on some $L^q$ space. Moreover, if every hyperbolic group is residually finite, then every convex-cocompact subgroup of $MCG(\Sigma _{g,p})$ is separable.

The aforementioned results follow from general theorems about composite rotating families, in the sense of [13], that come from a collection of subgroups of vertex stabilizers for the action of a group G on a hyperbolic graph X. We give conditions ensuring that the graph X/N is again hyperbolic and various properties of the action of G on X persist for the action of G/N on X/N.

We prove that viscosity solutions to fully nonlinear elliptic equations with degeneracy of double phase type are locally C1, γ-regular.

We prove certain L2(ℝn) bilinear estimates for Fourier extension operators associated to spheres and hyperboloids under the action of the k-plane transform. As the estimates are L2-based, they follow from bilinear identities: in particular, these are the analogues of a known identity for paraboloids, and may be seen as higher-dimensional versions of the classical L2(ℝ2)-bilinear identity for Fourier extension operators associated to curves in ℝ2.

We solve the inverse differential Galois problem over differential fields with a large field of constants of infinite transcendence degree over $\mathbb{Q}$. More generally, we show that over such a field, every split differential embedding problem can be solved. In particular, we solve the inverse differential Galois problem and all split differential embedding problems over $\mathbb{Q}_{p}(x)$.

We are concerned with a nonnegative solution to the scalar field equation

$$\Delta u + f(u) = 0{\rm in }{\open R}^N,\quad \mathop {\lim }\limits_{|x|\to \infty } u(x) = 0.$$

$f \in L^1_{loc}(\mathbb {R}) \cap \mathrm {Lip}_{loc}(0,\infty )$

$\int _0^u f(s)\,{\rm d}s < 0$

$u>0.$

$f \in \mathrm {Lip}_{loc}(0,\infty ).$

$f \in L^1_{loc}(\mathbb {R}) \cap C(0,\infty )$

Denote by m(G) the largest size of a minimal generating set of a finite group G. We estimate m(G) in terms of $\sum _{p\in \pi (G)}d_p(G),$ where we are denoting by dp(G) the minimal number of generators of a Sylow p-subgroup of G and by π(G) the set of prime numbers dividing the order of G.

In this paper, we prove a conjecture of Wei Zhang on comparison of certain local relative characters from which we draw some consequences for the Ichino–Ikeda conjecture for unitary groups.

We prove that the category of (rigidified) Breuil–Kisin–Fargues modules up to isogeny is Tannakian. We then introduce and classify Breuil–Kisin–Fargues modules with complex multiplication mimicking the classical theory for rational Hodge structures. In particular, we compute an avatar of a ‘$p$-adic Serre group’.

This is the second part of our study on the spatially heterogeneous predator–prey model where the interaction is governed by a Crowley–Martin type functional response. In part I, we have proved that when the predator competition is strong (i.e. k is large), the model has at most one positive steady-state solution for any $\mu \in \mathbb {R}$, moreover it is globally asymptotically stable for any $\mu >0$. This part is denoted to study the effect of saturation. Our result shows that the large saturation coefficient (i.e. large m) can not only lead to the uniqueness of positive solutions, but also lead to the multiplicity of positive solutions, moreover the stability of the corresponding positive solutions is also completely obtained. This work can be regarded as a supplement of Ref. [10].