The present article is concerned with the Lyapunov stability of stationary solutions to the Allen–Cahn equation with a strong irreversibility constraint, which was first intensively studied in [2] and can be reduced to an evolutionary variational inequality of obstacle type. As a feature of the obstacle problem, the set of stationary solutions always includes accumulation points, and hence, it is rather delicate to determine the stability of such non-isolated equilibria. Furthermore, the strongly irreversible Allen–Cahn equation can also be regarded as a (generalized) gradient flow; however, standard techniques for gradient flows such as linearization and Łojasiewicz–Simon gradient inequalities are not available for determining the stability of stationary solutions to the strongly irreversible Allen–Cahn equation due to the non-smooth nature of the obstacle problem.

Given a group G acting faithfully on a set S, we characterize precisely when the twisted Brin–Thompson group SVG is finitely presented. The answer is that SVG is finitely presented if and only if we have the following: G is finitely presented, the action of G on S has finitely many orbits of two-element subsets of S, and the stabilizer in G of any element of S is finitely generated. Since twisted Brin–Thompson groups are simple, a consequence is that any subgroup of a group admitting an action as above satisfies the Boone–Higman conjecture. In the course of proving this, we also establish a sufficient condition for a group acting cocompactly on a simply connected complex to be finitely presented, even if certain edge stabilizers are not finitely generated, which may be of independent interest.

Nilpotency concepts for skew braces are among the main tools with which we are nowadays classifying certain special solutions of the Yang–Baxter equation, a consistency equation that plays a relevant role in quantum statistical mechanics and in many areas of mathematics. In this context, two relevant questions have been raised in F. Cedó, A. Smoktunowicz and L. Vendramin (Skew left braces of nilpotent type. Proc. Lond. Math. Soc. (3) 118 (2019), 1367–1392) (see questions 2.34 and 2.35) concerning right- and central nilpotency. The aim of this short note is to give a negative answer to both questions: thus, we show that a finite strong-nil brace B need not be right-nilpotent. On a positive note, we show that there is one (and only one, by our examples) special case of the previous questions that actually holds. In fact, we show that if B is a skew brace of nilpotent type and $b\ \ast \ b=0$ for all $b\in B$, then B is centrally nilpotent.

We study an epidemic patch model that describes the disease spread in population with variable latency due to the differences in immunologic tolerance between individuals. We focus on whether the disease can spread in space that leads to the emergence of epidemic wave, that is the travelling wave solution with constant speed. We first establish some properties of the linearized wave profile equations, which are helpful in obtaining the priori estimates of travelling waves and wave speeds. Then, applying the truncation method and limiting arguments, we can obtain threshold propagation dynamics of the epidemic model. Our result gives a complete characterization of the existence, nonexistence and minimal wave speed of travelling waves. To the best of our knowledge, this is the first time to characterize the propagation dynamics of epidemic patch model with variable latency, which contributes to the understanding of the transmission phenomenon of disease.

We study the problem of extending an order-preserving real-valued Lipschitz map defined on a subset of a partially ordered metric space without increasing its Lipschitz constant and preserving its monotonicity. We show that a certain type of relation between the metric and order of the space, which we call radiality, is necessary and sufficient for such an extension to exist. Radiality is automatically satisfied by the equality relation, so the classical McShane–Whitney extension theorem is a special case of our main characterization result. As applications, we obtain a similar generalization of McShane’s uniformly continuous extension theorem, along with some functional representation results for radial partial orders.

We denote by $\mathcal{H}_{d,g,r}$ the Hilbert scheme of smooth curves, which is the union of components whose general point corresponds to a smooth, irreducible, and non-degenerate curve of degree d and genus g in $\mathbb{P}^r.$ In this article, we study $\mathcal{H}_{15,g,5}$ for every possible genus g and determine when it is irreducible. We also study the moduli map $\mathcal{H}_{15,g,5}\rightarrow\mathcal{M}_g$ and several key properties such as gonality of a general element as well as characterizing smooth elements of each component.

Several structural results about permutation groups of finite rank definable in differentially closed fields of characteristic zero (and other similar theories) are obtained. In particular, it is shown that every finite rank definably primitive permutation group is definably isomorphic to an algebraic permutation group living in the constants. Applications include the verification, in differentially closed fields, of the finite Morley rank permutation group conjectures of Borovik-Deloro and Borovik-Cherlin. Applying the results to binding groups for internality to the constants, it is deduced that if complete types p and q are of rank m and n, respectively, and are nonorthogonal, then the $(m+3)$rd Morley power of p is not weakly orthogonal to the $(n+3)$rd Morley power of q. An application to transcendence of generic solutions of pairs of algebraic differential equations is given.

We introduce two new notions called the Daugavet constant and Δ-constant of a point, which measure quantitatively how far the point is from being Daugavet point and Δ-point and allow us to study Daugavet and Δ-points in Banach spaces from a quantitative viewpoint. We show that these notions can be viewed as a localized version of certain global estimations of Daugavet and diametral local diameter two properties such as Daugavet indices of thickness. As an intriguing example, we present the existence of a Banach space X in which all points on the unit sphere have positive Daugavet constants despite the Daugavet indices of thickness of X being zero. Moreover, using the Daugavet and Δ-constants of points in the unit sphere, we describe the existence of almost Daugavet and Δ-points, as well as the set of denting points of the unit ball. We also present exact values of the Daugavet and Δ-constant on several classical Banach spaces, as well as Lipschitz-free spaces. In particular, it is shown that there is a Lipschitz-free space with a Δ-point, which is the furthest away from being a Daugavet point. Finally, we provide some related stability results concerning the Daugavet and Δ-constant.

This paper is concerned with a singular limit of the Kobayashi–Warren–Carter system, a phase field system modelling the evolutions of structures of grains. Under a suitable scaling, the limit system is formally derived when the interface thickness parameter tends to zero. Different from many other problems, it turns out that the limit system is a system involving fractional time derivatives, although the original system is a simple gradient flow. A rigorous derivation is given when the problem is reduced to a gradient flow of a single-well Modica–Mortola functional in a one-dimensional setting.

In this article, we calculate the Birkhoff spectrum in terms of the Hausdorff dimension of level sets for Birkhoff averages of continuous potentials for a certain family of diagonally affine iterated function systems. Also, we study Besicovitch–Eggleston sets for finite generalized Lüroth series number systems with redundancy. The redundancy refers to the fact that each number $x \in [0,1]$ has uncountably many expansions in the system. We determine the Hausdorff dimension of digit frequency sets for such expansions along fibres.

In this article, we study the following Schrödinger equation

\begin{align*}\begin{cases}-\Delta u -\frac{\mu}{|x|^2} u+\lambda u =f(u), &\text{in}~ \mathbb{R}^N\backslash\{0\},\\\int_{\mathbb{R}^{N}}|u|^{2}\mathrm{d} x=a, & u\in H^1(\mathbb{R}^{N}),\end{cases}\end{align*}

where $N\geq 3$, a > 0, and $\mu \lt \frac{(N-2)^2}{4}$. Here $\frac{1}{|x|^2} $ represents the Hardy potential (or ‘inverse-square potential’), λ is a Lagrange multiplier, and the nonlinearity function f satisfies the general Sobolev critical growth condition. Our main goal is to demonstrate the existence of normalized ground state solutions for this equation when $0 \lt \mu \lt \frac{(N-2)^2}{4}$. We also analyse the behaviour of solutions as $\mu\to0^+$ and derive the existence of normalized ground state solutions for the limiting case where µ = 0. Finally, we investigate the existence of normalized solutions when µ < 0 and analyse the asymptotic behaviour of solutions as $\mu\to 0^-$.

It is known that hyperbolic linear delay difference equations are shadowable on the half-line. In this article, we prove the converse and hence the equivalence between hyperbolicity and the positive shadowing property for the following two classes of linear delay difference equations: (a) for non-autonomous equations with finite delays and uniformly bounded compact coefficient operators in Banach spaces and (b) for Volterra difference equations with infinite delay in finite dimensional spaces.

The article considers systems of interacting particles on networks with adaptively coupled dynamics. Such processes appear frequently in natural processes and applications. Relying on the notion of graph convergence, we prove that for large systems the dynamics can be approximated by the corresponding continuum limit. Well-posedness of the latter is also established.