In this article, we deal with non-existence results, i.e., Liouville type results, for positive radial solutions of quasilinear elliptic equations with weights both in the entire $\mathbb R^N$ and in a ball, in the latter case under Dirichlet boundary conditions. The presence of weights, possibly singular or degenerate, makes the study fairly delicate. The proofs use a Pohozaev type identity combined with an accurate qualitative analysis of solutions. In the last part of the article, a non-existence theorem is proved for a Dirichlet problem with a convection term.

We provide a characterization of equivariant Fock covariant injective representations for product systems. We show that this characterization coincides with Nica covariance for compactly aligned product systems over right least common multiple semigroups of Kwaśniewski and Larsen and with the Toeplitz representations of a discrete monoid of Laca and Sehnem. By combining with the framework established by Katsoulis and Ramsey, we resolve the reduced Hao–Ng isomorphism problem for generalized gauge actions by discrete groups.

Asymptotic homogenisation is considered for problems with integral constraints imposed on a slowly varying microstructure; an insulator with an array of perfectly dielectric inclusions of slowly varying size serves as a paradigm. Although it is well-known how to handle each of these effects (integral constraints, slowly varying microstructure) independently within multiple scales analysis, additional care is needed when they are combined. Using the flux transport theorem, the multiple scales form of an integral constraint on a slowly varying domain is identified. The proposed form is applied to obtain a homogenised model for the electric potential in a dielectric composite, where the microstructure slowly varies and the integral constraint arises due to a statement of charge conservation. A comparison with multiple scales analysis of the problem with established approaches provides validation that the proposed form results in the correct homogenised model.

A p-arithmetic subgroup of $\mathbf {SL}_2(\mathbb {Q})$ like the Ihara group $\Gamma := \mathbf {SL}_2(\mathbb {Z}[1/p])$ acts by Möbius transformations on the Poincaré upper half plane $\mathcal H$ and on Drinfeld’s p-adic upper half plane ${\mathcal H_p := \mathbb {P}_1(\mathbb {C}_p)-\mathbb {P}_1(\mathbb {Q}_p)}$. The diagonal action of $\Gamma $ on the product is discrete, and the quotient $\Gamma \backslash (\mathcal H_p\times \mathcal H)$ can be envisaged as a ‘mock Hilbert modular surface’. According to a striking prediction of Neková$\check {\text {r}}$ and Scholl, the CM points on genuine Hilbert modular surfaces should give rise to ‘plectic Heegner points’ that encode nontrivial regulators attached, notably, to elliptic curves of rank two over real quadratic fields. This article develops the analogy between Hilbert modular surfaces and their mock counterparts, with the aim of transposing the plectic philosophy to the mock Hilbert setting, where the analogous plectic invariants are expected to lie in the alternating square of the Mordell–Weil group of certain elliptic curves of rank two over $\mathbb {Q}$.

We study the long-time asymptotics for the solution of the modified Camassa–Holm (mCH) equation with step-like initial data.

\begin{align*} &m_{t}+\left (m\left (u^{2}-u_{x}^{2}\right )\right )_{x}=0, \quad m=u-u_{xx}, \\[3pt] & {u(x,0)=u_0(x)\to \left \{ \begin{array}{l@{\quad}l} 1/c_+, &\ x\to +\infty, \\[3pt] 1/c_-, &\ x\to -\infty, \end{array}\right .} \end{align*}

$c_+$

$c_-$

$(y,t)$

$(\xi, c)$

$c=c_+/c_-$

$\{ (\xi, c)\,:\, \xi \in \mathbb{R}, \ c\gt 0, \ \xi =y/t\}$

$u(y,t)$

$\mathcal{O}(t^{-2})$

$\mathcal{O}(t^{-1})$

We consider the chemotaxis-Navier–Stokes system with generalised fluid dissipation in $\mathbb{R}^3$:

\begin{eqnarray*} \begin{cases} \partial _t n+u\cdot \nabla n=\Delta n- \nabla \cdot (\chi (c)n \nabla c),\\[5pt] \partial _t c+u \cdot \nabla c=\Delta c-nf(c),\\[5pt] \partial _t u +u \cdot \nabla u+\nabla P=-(\!-\Delta )^\alpha u-n\nabla \phi, \\[5pt] \nabla \cdot u=0, \end{cases} \end{eqnarray*}

$\alpha \gt \frac {3}{4}$

$\tilde {\textrm {a}}$

$\alpha \gt \frac {1}{2}$

$\alpha \geq \frac {5}{4}$

$\frac {3}{4}\lt \alpha \lt \frac {5}{4}$

$L^2$

We describe algebraically, diagrammatically, and in terms of weight vectors, the restriction of tensor powers of the standard representation of quantum $\mathfrak {sl}_2$ to a coideal subalgebra. We realize the category as a module category over the monoidal category of type $\pm 1$ representations in terms of string diagrams and via generators and relations. The idempotents projecting onto the quantized eigenspaces are described as type $B/D$ analogues of Jones–Wenzl projectors. As an application, we introduce and give recursive formulas for analogues of $\Theta$-networks.

In his last paper, William Thurston defined the Master Teapot as the closure of the set of pairs $(z,s)$, where s is the slope of a tent map $T_s$ with the turning point periodic, and the complex number z is a Galois conjugate of s. In this case $1/z$ is a zero of the kneading determinant of $T_s$. We remove the restriction that the turning point is periodic, and sometimes look beyond tent maps. However, we restrict our attention to zeros $x=1/z$ in the real interval $(0,1)$. By the results of Milnor and Thurston, the kneading determinant has such a zero if and only if the map has positive topological entropy. We show that the first (smallest) zero is simple, but among other zeros there may be multiple ones. We describe a class of unimodal maps, so-called R-even ones, whose kneading determinant has only one zero in $(0,1)$. In contrast, we show that generic mixing tent maps have kneading determinants with infinitely many zeros in $(0,1)$. We prove that the second zero in $(0,1)$ of the kneading determinant of a unimodal map, provided it exists, is always greater than or equal to $\sqrt [3]{1/2}$, and if the kneading sequence begins with $RL^NR$, $N\geq 2$, then the best lower bound for the second zero is in fact $\sqrt [N+1]{1/2}$. We also investigate (partially numerically) the shape of the Real Teapot, consisting of the pairs $(s,x)$, where x in $(0,1)$ is a zero of the kneading determinant of $T_s$, and $s\in (1,2]$.

Bioreactor scaffolds must be designed to facilitate adequate nutrient delivery to the growing tissue they support. For perfusion bioreactors, the dominant transport process is determined by the scale of fluid velocity relative to diffusion and the geometry of the scaffold. In this paper, models of nutrient transport in a fibrous bioreactor scaffold are developed using homogenisation via multiscale asymptotics. The scaffold is modelled as an ensemble of aligned strings surrounded by viscous, slowly flowing fluid. Multiple scales analysis is carried out for various parameter regimes which give rise to macroscale transport models that incorporate the effects of advection, reaction and diffusion. Multiple scales in both space and time are employed when macroscale advection balances macroscale diffusion. The microscale model is solved to obtain the effective diffusion coefficient and simple solutions to the macroscale problem are presented for each regime.

We show that the fundamental groups of smooth $4$-manifolds that admit geometric decompositions in the sense of Thurston have asymptotic dimension at most four, and equal to four when aspherical. We also show that closed $3$-manifold groups have asymptotic dimension at most three. Our proof method yields that the asymptotic dimension of closed $3$-dimensional Alexandrov spaces is at most three. Thus, we obtain that the Novikov conjecture holds for closed $4$-manifolds with such a geometric decomposition and for closed $3$-dimensional Alexandrov spaces. Consequences of these results include a vanishing result for the Yamabe invariant of certain $0$-surgered geometric $4$-manifolds and the existence of zero in the spectrum of aspherical smooth $4$-manifolds with a geometric decomposition.

We introduce and study the notion of hereditary frequent hypercyclicity, which is a reinforcement of the well-known concept of frequent hypercyclicity. This notion is useful for the study of the dynamical properties of direct sums of operators; in particular, a basic observation is that the direct sum of a hereditarily frequently hypercyclic operator with any frequently hypercyclic operator is frequently hypercyclic. Among other results, we show that operators satisfying the frequent hypercyclicity criterion are hereditarily frequently hypercyclic, as well as a large class of operators whose unimodular eigenvectors are spanning with respect to the Lebesgue measure. However, we exhibit two frequently hypercyclic weighted shifts $B_w,B_{w'}$ on $c_0(\mathbb {Z}_+)$ whose direct sum ${B_w\oplus B_{w'}}$ is not $\mathcal {U}$-frequently hypercyclic (so that neither of them is hereditarily frequently hypercyclic), and we construct a C-type operator on $\ell _p(\mathbb {Z}_+)$, $1\le p<\infty $, which is frequently hypercyclic but not hereditarily frequently hypercyclic. We also solve several problems concerning disjoint frequent hypercyclicity: we show that for every $N\in \mathbb {N}$, any disjoint frequently hypercyclic N-tuple of operators $(T_1,\ldots ,T_N)$ can be extended to a disjoint frequently hypercyclic $(N+1)$-tuple $(T_1,\ldots ,T_N, T_{N+1})$ as soon as the underlying space supports a hereditarily frequently hypercyclic operator; we construct a disjoint frequently hypercyclic pair which is not densely disjoint hypercyclic; and we show that the pair $(D,\tau _a)$ is disjoint frequently hypercyclic, where D is the derivation operator acting on the space of entire functions and $\tau _a$ is the operator of translation by $a\in \mathbb {C}\setminus \{ 0\}$. Part of our results are in fact obtained in the general setting of Furstenberg families.

Optimal transport tasks naturally arise in gas networks, which include a variety of constraints such as physical plausibility of the transport and the avoidance of extreme pressure fluctuations. To define feasible optimal transport plans, we utilize a $p$-Wasserstein metric and similar dynamic formulations minimizing the kinetic energy necessary for moving gas through the network, which we combine with suitable versions of Kirchhoff’s law as the coupling condition at nodes. In contrast to existing literature, we especially focus on the non-standard case $p \neq 2$ to derive an overdamped isothermal model for gases through $p$-Wasserstein gradient flows in order to uncover and analyze underlying dynamics. We introduce different options for modelling the gas network as an oriented graph including the possibility to store gas at interior vertices and to put in or take out gas at boundary vertices.

In this paper, we prove that if a three-dimensional quasi-projective variety X over an algebraically closed field of characteristic $p>3$ has only log canonical singularities, then so does a general hyperplane section H of X. We also show that the same is true for klt singularities, which is a slight extension of [15]. In the course of the proof, we provide a sufficient condition for log canonical (resp. klt) surface singularities to be geometrically log canonical (resp. geometrically klt) over a field.

In this paper, we consider an optimal distributed control problem for a reaction-diffusion-based SIR epidemic model with human behavioural effects. We develop a model wherein non-pharmaceutical intervention methods are implemented, but a portion of the population does not comply with them, and this non-compliance affects the spread of the disease. Drawing from social contagion theory, our model allows for the spread of non-compliance parallel to the spread of the disease. The quantities of interest for control are the reduction in infection rate among the compliant population, the rate of spread of non-compliance and the rate at which non-compliant individuals become compliant after, e.g., receiving more or better information about the underlying disease. We prove the existence of global-in-time solutions for fixed controls and study the regularity properties of the resulting control-to-state map. The existence of optimal control is then established in an abstract framework for a fairly general class of objective functions. Necessary first–order optimality conditions are obtained via a Lagrangian-based stationarity system. We conclude with a discussion regarding minimisation of the size of infected and non-compliant populations and present simulations with various parameters values to demonstrate the behaviour of the model.

We characterize hyperbolic groups in terms of quasigeodesics in the Cayley graph forming regular languages. We also obtain a quantitative characterization of hyperbolicity of geodesic metric spaces by the non-existence of certain local $(3,0)$-quasigeodesic loops. As an application, we make progress towards a question of Shapiro regarding groups admitting a uniquely geodesic Cayley graph.

Inspired by Nakamura’s work [36] on $\epsilon $-isomorphisms for $(\varphi ,\Gamma )$-modules over (relative) Robba rings with respect to the cyclotomic theory, we formulate an analogous conjecture for L-analytic Lubin-Tate $(\varphi _L,\Gamma _L)$-modules over (relative) Robba rings for any finite extension L of $\mathbb {Q}_p.$ In contrast to Kato’s and Nakamura’s setting, our conjecture involves L-analytic cohomology instead of continuous cohomology within the generalized Herr complex. Similarly, we restrict to the identity components of $D_{cris}$ and $D_{dR},$ respectively. For rank one modules of the above type or slightly more generally for trianguline ones, we construct $\epsilon $-isomorphisms for their Lubin-Tate deformations satisfying the desired interpolation property.

This paper is focused on the existence and uniqueness of nonconstant steady states in a reaction–diffusion–ODE system, which models the predator–prey interaction with Holling-II functional response. Firstly, we aim to study the occurrence of regular stationary solutions through the application of bifurcation theory. Subsequently, by a generalized mountain pass lemma, we successfully demonstrate the existence of steady states with jump discontinuity. Furthermore, the structure of stationary solutions within a one-dimensional domain is investigated and a variety of steady-state solutions are built, which may exhibit monotonicity or symmetry. In the end, we create heterogeneous equilibrium states close to a constant equilibrium state using bifurcation theory and examine their stability.

We study the existence and regularity of minimizers of the neo-Hookean energy in the closure of classes of deformations without cavitation. The exclusion of cavitation is imposed in the form of the divergence identities, which is equivalent to the well-known condition (INV) with $\operatorname{Det} = \operatorname{det}$. We show that the neo-Hookean energy admits minimizers in classes of maps that are one-to-one a.e. with positive Jacobians, provided that these maps are the weak limits of sequences of maps that satisfy the divergence identities. In particular, these classes include the weak closure of diffeomorphisms and the weak closure of homeomorphisms satisfying Lusin’s condition N. Moreover, if the minimizers satisfy condition (INV), then their inverses have Sobolev regularity. This extends a recent result by Doležalová, Hencl, and Molchanova by showing that the minimizers they obtained enjoy extra regularity properties and that the existence of minimizers can still be obtained even when their coercivity assumption is relaxed.