The dynamics of interfaces in slow diffusion equations with strong absorption are studied. Asymptotic methods are used to give descriptions of the behaviour local to a comprehensive range of possible singular events that can occur in any evolution. These events are: when an interface changes its direction of propagation (reversing and anti-reversing), when an interface detaches from an absorbing obstacle (detaching), when two interfaces are formed by film rupture (touchdown) and when the solution undergoes extinction. Our account of extinction and self-similar reversing and anti-reversing is built upon previous work; results on non-self-similar reversing and anti-reversing and on the various types of detachment and touchdown are developed from scratch. In all cases, verification of the asymptotic results against numerical solutions to the full PDE is provided. Self-similar solutions, both of the full equation and of its asymptotic limits, play a central role in the analysis.

In this paper, we consider the dynamical behaviour of a reaction–diffusion model for a population residing in a one-dimensional habit, with emphasis on the effects of boundary conditions and protection zone. We assume that the population is subjected to a strong Allee effect in its natural domain but obeys a monostable nonlinear growth in the protection zone $[L_1,\, L_2]$ with two constants satisfying $0\leq L_1< L_2$, and the general Robin condition is imposed on $x=0$ (i.e. $u(t,\,0)=bu_x(t,\,0)$ with $b\geq 0$). We show the existence of two critical values $0< L_*\leq L^*$, and prove that a vanishing–transition–spreading trichotomy result holds when the length of protection zone is smaller than $L_*$; a transition–spreading dichotomy result holds when the length of protection zone is between $L_*$ and $L^*$; only spreading happens when the length of protection zone is larger than $L^*$. Based on the properties of $L_*$, we obtain the precise strategies for an optimal protection zone: if $b$ is large (i.e. $b\geq 1/\sqrt {-g'(0)}$), the protection zone should start from somewhere near $0$; while if $b$ is small (i.e. $b< 1/\sqrt {-g'(0)}$), then the protection zone should start from somewhere away from $0$, and as far away from $0$ as possible.

We construct an anticyclotomic Euler system for the Rankin–Selberg convolutions of two modular forms, using p-adic families of generalised Gross–Kudla–Schoen diagonal cycles. As applications of this construction, we prove new results on the Bloch–Kato conjecture in analytic ranks zero and one, and a divisibility towards an Iwasawa main conjecture.

We study the problem of finding the root vertex in large growing networks. We prove that it is possible to construct confidence sets of size independent of the number of vertices in the network that contain the root vertex with high probability in various models of random networks. The models include uniform random recursive dags and uniform Cooper-Frieze random graphs.

We show that for a fixed $q$, the number of $q$-ary $t$-error correcting codes of length $n$ is at most $2^{(1 + o(1)) H_q(n,t)}$ for all $t \leq (1 - q^{-1})n - 2\sqrt{n \log n}$, where $H_q(n, t) = q^n/ V_q(n,t)$ is the Hamming bound and $V_q(n,t)$ is the cardinality of the radius $t$ Hamming ball. This proves a conjecture of Balogh, Treglown, and Wagner, who showed the result for $t = o(n^{1/3} (\log n)^{-2/3})$.

We extend a recent argument of Kahn, Narayanan and Park ((2021) Proceedings of the AMS 149 3201–3208) about the threshold for the appearance of the square of a Hamilton cycle to other spanning structures. In particular, for any spanning graph, we give a sufficient condition under which we may determine its threshold. As an application, we find the threshold for a set of cyclically ordered copies of $C_4$ that span the entire vertex set, so that any two consecutive copies overlap in exactly one edge and all overlapping edges are disjoint. This answers a question of Frieze. We also determine the threshold for edge-overlapping spanning $K_r$-cycles.

We prove that if a unimodular random graph is almost surely planar and has finite expected degree, then it has a combinatorial embedding into the plane which is also unimodular. This implies the claim in the title immediately by a theorem of Angel, Hutchcroft, Nachmias and Ray [2]. Our unimodular embedding also implies that all the dichotomy results of [2] about unimodular maps extend in the one-ended case to unimodular random planar graphs.

Alweiss, Lovett, Wu, and Zhang introduced $q$-spread hypergraphs in their breakthrough work regarding the sunflower conjecture, and since then $q$-spread hypergraphs have been used to give short proofs of several outstanding problems in probabilistic combinatorics. A variant of $q$-spread hypergraphs was implicitly used by Kahn, Narayanan, and Park to determine the threshold for when a square of a Hamiltonian cycle appears in the random graph $G_{n,p}$. In this paper, we give a common generalization of the original notion of $q$-spread hypergraphs and the variant used by Kahn, Narayanan, and Park.

We analyse the $\Gamma$-convergence of general non-local convolution type functionals with varying densities depending on the space variable and on the symmetrized gradient. The limit is a local free-discontinuity functional, where the bulk term can be completely characterized in terms of an asymptotic cell formula. From that, we can deduce an homogenisation result in the stochastic setting.

We introduce a category of filtered F-isocrystals and construct a symbol map from Milnor K-theory to the group of 1-extensions of filtered F-isocrystals. We show that our symbol map is compatible with the syntomic symbol map to the log syntomic cohomology by Kato and Tsuji. These are fundamental materials in our computations of syntomic regulators which we work in other papers.

We prove that real topological Hochschild homology $\mathrm {THR}$ for schemes with involution satisfies base change and descent for the ${\mathbb {Z}/2}$-isovariant étale topology. As an application, we provide computations for the projective line (with and without involution) and the higher-dimensional projective spaces.

Stability is among the most important concepts in dynamical systems. Local stability is well-studied, whereas determining the ‘global stability’ of a nonlinear system is very challenging. Over the last few decades, many different ideas have been developed to address this issue, primarily driven by concrete applications. In particular, several disciplines suggested a web of concepts under the headline ‘resilience’. Unfortunately, there are many different variants and explanations of resilience, and often, the definitions are left relatively vague, sometimes even deliberately. Yet, to allow for a structural development of a mathematical theory of resilience that can be used across different areas, one has to ensure precise starting definitions and provide a mathematical comparison of different resilience measures. In this work, we provide a systematic review of the most relevant indicators of resilience in the context of continuous dynamical systems, grouped according to their mathematical features. The indicators are also generalised to be applicable to any attractor. These steps are important to ensure a more reliable, quantitatively comparable and reproducible study of resilience in dynamical systems. Furthermore, we also develop a new concept of resilience against certain nonautonomous perturbations to demonstrate how one can naturally extend our framework. All the indicators are finally compared via the analysis of a classic scalar model from population dynamics to show that direct quantitative application-based comparisons are an immediate consequence of a detailed mathematical analysis.

We develop a geometric method to establish the existence and uniqueness of equilibrium states associated to some Hölder potentials for center isometries (as are regular elements of Anosov actions), in particular, the entropy maximizing measure and the SRB measure. A characterization of equilibrium states in terms of their disintegrations along stable and unstable foliations is also given. Finally, we show that the resulting system is isomorphic to a Bernoulli scheme.

Which patterns must a two-colouring of $K_n$ contain if each vertex has at least $\varepsilon n$ red and $\varepsilon n$ blue neighbours? We show that when $\varepsilon \gt 1/4$, $K_n$ must contain a complete subgraph on $\Omega (\log n)$ vertices where one of the colours forms a balanced complete bipartite graph.

When $\varepsilon \leq 1/4$, this statement is no longer true, as evidenced by the following colouring $\chi$ of $K_n$. Divide the vertex set into $4$ parts nearly equal in size as $V_1,V_2,V_3, V_4$, and let the blue colour class consist of the edges between $(V_1,V_2)$, $(V_2,V_3)$, $(V_3,V_4)$, and the edges contained inside $V_2$ and inside $V_3$. Surprisingly, we find that this obstruction is unique in the following sense. Any two-colouring of $K_n$ in which each vertex has at least $\varepsilon n$ red and $\varepsilon n$ blue neighbours (with $\varepsilon \gt 0$) contains a vertex set $S$ of order $\Omega _{\varepsilon }(\log n)$ on which one colour class forms a balanced complete bipartite graph, or which has the same colouring as $\chi$.

We use new methods, specific for non-locally convex quasi-Banach spaces, to investigate when the quasi-greedy bases of a $p$-Banach space for $0< p<1$ are democratic. The novel techniques we obtain permit to show in particular that all quasi-greedy bases of the Hardy space $H_p({\mathbb {D}})$ for $0< p<1$ are democratic while, in contrast, no quasi-greedy basis of $H_p({\mathbb {D}}^d)$ for $d\ge 2$ is, solving thus a problem that was raised in [7]. Applications of our results to other spaces of interest both in functional analysis and approximation theory are also provided.