In this paper, we consider reaction-diffusion epidemic models with mass action or standard incidence mechanism and study the impact of limiting population movement on disease transmissions. We set either the dispersal rate of the susceptible or infected people to zero and study the corresponding degenerate reaction-diffusion model. Our main approach to study the global dynamics of these models is to construct delicate Lyapunov functions. Our results show that the consequences of limiting the movement of susceptible or infected people depend on transmission mechanisms, model parameters and population size.

We prove new results concerning the additive Galois module structure of wildly ramified non-abelian extensions $K/\mathbb{Q}$ with Galois group isomorphic to $A_4$, $S_4$, $A_5$, and dihedral groups of order $2p^n$ for certain prime powers $p^n$. In particular, when $K/\mathbb{Q}$ is a Galois extension with Galois group $G$ isomorphic to $A_4$, $S_4$ or $A_5$, we give necessary and sufficient conditions for the ring of integers $\mathcal{O}_{K}$ to be free over its associated order in the rational group algebra $\mathbb{Q}[G]$.

For $s_1,\,s_2\in (0,\,1)$ and $p,\,q \in (1,\, \infty )$, we study the following nonlinear Dirichlet eigenvalue problem with parameters $\alpha,\, \beta \in \mathbb {R}$ driven by the sum of two nonlocal operators:

\[ (-\Delta)^{s_1}_p u+(-\Delta)^{s_2}_q u=\alpha|u|^{p-2}u+\beta|u|^{q-2}u\ \text{in }\Omega, \quad u=0\ \text{in } \mathbb{R}^d \setminus \Omega, \quad \mathrm{(P)} \]

$\Omega \subset \mathbb {R}^d$

$\alpha,\,\beta$

$(\alpha,\, \beta )$

$p$

$q$

This article is devoted to the analysis of the parabolic–parabolic chemotaxis system with multi-components over $\mathbb{R}^2$. The optimal small initial condition on the global existence of solutions for multi-species chemotaxis model in the fully parabolic situation had not been attained as far as the author knows. In this paper, we prove that under the sub-critical mass condition, any solutions to conflict-free system exist globally. Moreover, the global existence of solutions to system with strong self-repelling effect has been discussed even for large initial data. The proof is based on the modified free energy functional and the Moser–Trudinger inequality for system.

We examine several extensions to the basic Susceptible-Infected-Recovered model, which are able to induce recurrent outbreaks (the basic Susceptible-Infected-Recovered model by itself does not exhibit recurrent outbreaks). We first analyse how slow seasonal variations can destabilise the endemic equilibrium, leading to recurrent outbreaks. In the limit of slow immunity loss, we derive asymptotic thresholds that characterise this transition. In the outbreak regime, we use asymptotic matching to obtain a two-dimensional discrete map which describes outbreak times and strength. We then analyse the resulting map using linear stability and numerics. As the frequency of forcing is increased, the map exhibits a period-doubling route to chaos which alternates with periodic outbreaks of increasing frequency. Other extensions that can lead to recurrent outbreaks include the addition of noise, state-dependent variation and fine-graining of model classes.

Every Thurston map $f\colon S^2\rightarrow S^2$ on a $2$-sphere $S^2$ induces a pull-back operation on Jordan curves $\alpha \subset S^2\smallsetminus {P_f}$, where ${P_f}$ is the postcritical set of f. Here the isotopy class $[f^{-1}(\alpha )]$ (relative to ${P_f}$) only depends on the isotopy class $[\alpha ]$. We study this operation for Thurston maps with four postcritical points. In this case, a Thurston obstruction for the map f can be seen as a fixed point of the pull-back operation. We show that if a Thurston map f with a hyperbolic orbifold and four postcritical points has a Thurston obstruction, then one can ‘blow up’ suitable arcs in the underlying $2$-sphere and construct a new Thurston map $\widehat f$ for which this obstruction is eliminated. We prove that no other obstruction arises and so $\widehat f$ is realized by a rational map. In particular, this allows for the combinatorial construction of a large class of rational Thurston maps with four postcritical points. We also study the dynamics of the pull-back operation under iteration. We exhibit a subclass of our rational Thurston maps with four postcritical points for which we can give positive answer to the global curve attractor problem.

Inspired by a result in T. H. Colding. (16). Acta. Math. 209(2) (2012), 229-263 [16] of Colding, the present paper studies the Green function $G$ on a non-parabolic $\operatorname {RCD}(0,\,N)$ space $(X,\, \mathsf {d},\, \mathfrak {m})$ for some finite $N>2$. Defining $\mathsf {b}_x=G(x,\, \cdot )^{\frac {1}{2-N}}$ for a point $x \in X$, which plays a role of a smoothed distance function from $x$, we prove that the gradient $|\nabla \mathsf {b}_x|$ has the canonical pointwise representative with the sharp upper bound in terms of the $N$-volume density $\nu _x=\lim _{r\to 0^+}\frac {\mathfrak {m} (B_r(x))}{r^N}$ of $\mathfrak {m}$ at $x$;

\[ |\nabla \mathsf{b}_x|(y) \le \left(N(N-2)\nu_x\right)^{\frac{1}{N-2}}, \quad \text{for any }y \in X \setminus \{x\}. \]

$y \in X \setminus \{x\}$

$N$

$\operatorname {RCD}(N-2,\, N-1)$

$x$

$N$

$N$

$\mathbb {R}^N$

$\operatorname {RCD}(0,\,N)$

We establish a generalized Rieffel correspondence for ideals in equivalent Fell bundles.

In this paper, we classify simple smooth modules over the mirror Heisenberg–Virasoro algebra ${\mathfrak {D}}$, and simple smooth modules over the twisted Heisenberg–Virasoro algebra $\bar {\mathfrak {D}}$ with non-zero level. To this end we generalize Sugawara operators to smooth modules over the Heisenberg algebra, and develop new techniques. As applications, we characterize simple Whittaker modules and simple highest weight modules over ${\mathfrak {D}}$. A vertex-algebraic interpretation of our result is the classification of simple weak twisted and untwisted modules over the Heisenberg–Virasoro vertex algebras. We also present a few examples of simple smooth ${\mathfrak {D}}$-modules and $\bar {\mathfrak {D}}$-modules induced from simple modules over finite dimensional solvable Lie algebras, that are not tensor product modules of Virasoro modules and Heisenberg modules. This is very different from the case of simple highest weight modules over $\mathfrak {D}$ and $\bar {\mathfrak {D}}$ which are always tensor products of simple Virasoro modules and simple Heisenberg modules.

Wigner’s theorem characterizes isometries of the set of all rank one projections on a Hilbert space. In metric geometry, nonexpansive maps and noncontractive maps are well-studied generalizations of isometries. We show that under certain conditions Wigner symmetries can be characterized as nonexpansive or noncontractive maps on the set of all projections of rank one. The assumptions required for such characterizations are injectivity or surjectivity and they differ in the finite and the infinite-dimensional case. Motivated by a recently obtained optimal version of Uhlhorn’s generalization of Wigner’s theorem, we also give a description of nonexpansive maps which satisfy a condition that is much weaker than surjectivity. Such maps do not need to be Wigner symmetries. The optimality of all presented results is shown by counterexamples.

This paper examines the issue of derivative pricing within the framework of a fractional stochastic volatility model. We present a deterministic partial differential equation system to derive an approximate expression for the derivative price. The proposed approach allows for the stochastic volatility to be expressed as a composition of deterministic functions of time and a fractional Ornstein–Uhlenbeck process. We apply this method to the European option pricing under the fractional Stein–Stein volatility model, demonstrating its feasibility and reliability through numerical simulations. Our numerical simulations also illustrate the impact of the parameters in the fractional stochastic volatility model on the option price.

Let H be the Hermite operator $-\Delta +|x|^2$ on $\mathbb {R}^n$. We prove a weighted $L^2$ estimate of the maximal commutator operator $\sup _{R>0}|[b, S_R^\lambda (H)](f)|$, where $ [b, S_R^\lambda (H)](f) = bS_R^\lambda (H) f - S_R^\lambda (H)(bf) $ is the commutator of a BMO function b and the Bochner–Riesz means $S_R^\lambda (H)$ for the Hermite operator H. As an application, we obtain the almost everywhere convergence of $[b, S_R^\lambda (H)](f)$ for large $\lambda $ and $f\in L^p(\mathbb {R}^n)$.

A nonlinear evolution equation correct to fourth order is developed for gravity-capillary waves on linear shear currents in finite water depth. Therefore, this equation covers both effects of depth uniform currents and uniform vorticity. Starting from this equation, an instability analysis is then made for narrow banded uniform Stokes waves. The notable feature is that our investigation due to fourth order shows a remarkable improvement compared with the third-order one, and produces an excellent result compatible with the exact result of Longuet-Higgins. We observe that linear shear currents considerably change the modulational instability properties of capillary-gravity waves, such as the growth rate and bandwidth of instability.

We present a unified approach to the processes of inversion and duality for quasilinear and $1$-quasilinear maps; in particular, for centralizers and differentials generated by interpolation methods.

The algebraic mapping torus $M_{\Phi }$ of a group $G$ with an automorphism $\Phi$ is the HNN-extension of $G$ in which conjugation by the stable letter performs $\Phi$. We classify the Dehn functions of $M_{\Phi }$ in terms of $\Phi$ for a number of right-angled Artin groups (RAAGs) $G$, including all $3$-generator RAAGs and $F_k \times F_l$ for all $k,l \geq 2$.

Given a group $G$ and an integer $n\geq 0$, we consider the family ${\mathcal F}_n$ of all virtually abelian subgroups of $G$ of $\textrm{rank}$ at most $n$. In this article, we prove that for each $n\ge 2$ the Bredon cohomology, with respect to the family ${\mathcal F}_n$, of a free abelian group with $\textrm{rank}$ $k \gt n$ is nontrivial in dimension $k+n$; this answers a question of Corob Cook et al. (Homology Homotopy Appl. 19(2) (2017), 83–87, Question 2.7). As an application, we compute the minimal dimension of a classifying space for the family ${\mathcal F}_n$ for braid groups, right-angled Artin groups, and graphs of groups whose vertex groups are infinite finitely generated virtually abelian groups, for all $n\ge 2$. The main tools that we use are the Mayer–Vietoris sequence for Bredon cohomology, Bass–Serre theory, and the Lück–Weiermann construction.

We investigate the existence and branching patterns of wave trains in the mass-in-mass (MiM) lattice, which is a variant of the Fermi–Pasta–Ulam (FPU) lattice. In contrast to FPU lattice, we have to solve coupled advance-delay differential equations, which are reduced to a finite-dimensional bifurcation equation with an inherited Hamiltonian structure by applying a Lyapunov–Schmidt reduction and invariant theory. We establish a link between the MiM lattice and the monatomic FPU lattice. That is, the monochromatic and bichromatic wave trains persist near $\mu =0$ in the nonresonance case and in the resonance case $p:q$ where $q$ is not an integer multiple of $p$. Furthermore, we obtain the multiplicity of bichromatic wave trains in $p:q$ resonance where $q$ is an integer multiple of $p$, based on the singular theorem.