In this paper, we consider a delayed discrete single population patch model in advective environments. The individuals are subject to both random and directed movements, and there is a net loss of individuals at the downstream end due to the flow into a lake. Choosing time delay as a bifurcation parameter, we show the existence of Hopf bifurcations for the model. In homogeneous non-advective environments, it is well known that the first Hopf bifurcation value is independent of the dispersal rate. In contrast, for homogeneous advective environments, the first Hopf bifurcation value depends on the dispersal rate. Moreover, we show that the first Hopf bifurcation value in advective environments is larger than that in non-advective environments if the dispersal rate is large or small, which suggests that directed movements of the individuals inhibit the occurrence of Hopf bifurcations.

We compute the Jantzen filtration of a $\mathcal {D}$-module on the flag variety of $\operatorname {\mathrm {SL}}_2(\mathbb {C})$. At each step in the computation, we illustrate the $\mathfrak {sl}_2(\mathbb {C})$-module structure on global sections to give an algebraic picture of this geometric computation. We conclude by showing that the Jantzen filtration on the $\mathcal {D}$-module agrees with the algebraic Jantzen filtration on its global sections, demonstrating a famous theorem of Beilinson and Bernstein.

Let $X=GC$ be a group, where C is a cyclic group and G is either a generalized quaternion group or a dihedral group such that $C\cap G=1$. In this paper, X is characterized and, moreover, a complete classification for $X$ is given, provided that G is a generalized quaternion group and C is core-free.

We investigate the mean-field dynamics of stochastic McKean differential equations with heterogeneous particle interactions described by large network structures. To express a wide range of graphs, from dense to sparse structures, we incorporate the recently developed graph limit theory of graphops into the limiting McKean–Vlasov equations. Global stability of the splay steady state is proven via a generalised entropy method, leading to explicit graph structure-dependent decay rates. We highlight the robustness of the entropy approach by extending the results to the closely related Sakaguchi–Kuramoto model with intrinsic frequency distributions. We also present central examples of random graphs, such as power law graphs and the spherical graphop, and analyse the limitations of the applied methodology.

We introduce a free boundary model to study the effect of vesicle transport onto neurite growth. It consists of systems of drift-diffusion equations describing the evolution of the density of antero- and retrograde vesicles in each neurite coupled to reservoirs located at the soma and the growth cones of the neurites, respectively. The model allows for a change of neurite length as a function of the vesicle concentration in the growth cones. After establishing existence and uniqueness for the time-dependent problem, we briefly comment on possible types of stationary solutions. Finally, we provide numerical studies on biologically relevant scales using a finite volume scheme. We illustrate the capability of the model to reproduce cycles of extension and retraction.

We give a generators-and-relations description of the reduced versions of quiver quantum toroidal algebras, which act on the spaces of BPS states associated to (noncompact) toric Calabi–Yau threefolds X. As an application, we obtain a description of the K-theoretic Hall algebra of (the quiver with potential associated to) X, modulo torsion.

In this article, we study rational matrix representations of VZ p-groups (p is any prime). Using our findings on VZ p-groups, we explicitly obtain all inequivalent irreducible rational matrix representations of all p-groups of order $\leq p^4$. Furthermore, we establish combinatorial formulae to determine the Wedderburn decompositions of rational group algebras for VZ p-groups and all p-groups of order $\leq p^4$, ensuring simplicity in the process.

For $E \subset \mathbb {N}$, a subset $R \subset \mathbb {N}$ is E-intersective if for every $A \subset E$ having positive relative density, $R \cap (A - A) \neq \varnothing $. We say that R is chromatically E-intersective if for every finite partition $E=\bigcup _{i=1}^k E_i$, there exists i such that $R\cap (E_i-E_i)\neq \varnothing $. When $E=\mathbb {N}$, we recover the usual notions of intersectivity and chromatic intersectivity. We investigate to what extent the known intersectivity results hold in the relative setting when $E = \mathbb {P}$, the set of primes, or other sparse subsets of $\mathbb {N}$. Among other things, we prove the following: (1) the set of shifted Chen primes $\mathbb {P}_{\mathrm {Chen}} + 1$ is both intersective and $\mathbb {P}$-intersective; (2) there exists an intersective set that is not $\mathbb {P}$-intersective; (3) every $\mathbb {P}$-intersective set is intersective; (4) there exists a chromatically $\mathbb {P}$-intersective set which is not intersective (and therefore not $\mathbb {P}$-intersective).

For given $\epsilon>0$ and $b\in \mathbb {R}^m$, we say that a real $m\times n$ matrix A is $\epsilon $-badly approximable for the target b if

$$ \begin{align*}\liminf_{q\in\mathbb{Z}^n, \|q\|\to\infty} \|q\|^n \langle Aq-b\rangle^m \geq \epsilon,\end{align*} $$

$\langle \cdot \rangle $

$\epsilon $

$\epsilon $

$\epsilon $

The pth ($p\geq 1$) moment exponential stability, almost surely exponential stability and stability in distribution for stochastic McKean–Vlasov equation are derived based on some distribution-dependent Lyapunov function techniques.

We prove an effective estimate with a power saving error term for the number of square-tiled surfaces in a connected component of a stratum of quadratic differentials whose vertical and horizontal foliations belong to prescribed mapping class group orbits and which have at most L squares. This result strengthens asymptotic counting formulas in the work of Delecroix, Goujard, Zograf, Zorich, and the author.

This article studies the dynamical behaviour of classical solutions of a hyperbolic system of balance laws, derived from a chemotaxis model with logarithmic sensitivity, with time-dependent boundary conditions. It is shown that under suitable assumptions on the boundary data, solutions starting in the $H^2$-space exist globally in time and the differences between the solutions and their corresponding boundary data converge to zero as time goes to infinity. There is no smallness restriction on the magnitude of the initial perturbations. Moreover, numerical simulations show that the assumptions on the boundary data are necessary for the above-mentioned results to hold true. In addition, numerical results indicate that the solutions converge asymptotically to time-periodic states if the boundary data are time-periodic.