Recently, the Kac formula for the conditional expectation of the first recurrence time of a conditionally ergodic conditional expectation preserving system was established in the measure-free setting of vector lattices (Riesz spaces). We now give a formulation of the Kakutani–Rokhlin decomposition for conditionally ergodic systems in terms of components of weak order units in a vector lattice. In addition, we prove that every aperiodic conditional expectation preserving system can be approximated by a periodic system.

We define the chain Sobolev space on a possibly non-complete metric measure space in terms of chain upper gradients. In this context, ɛ-chains are finite collections of points with distance at most ɛ between consecutive points. They play the role of discrete curves. Chain upper gradients are defined accordingly and the chain Sobolev space is defined by letting the size parameter ɛ going to zero. In the complete setting, we prove that the chain Sobolev space is equal to the classical notions of Sobolev spaces in terms of relaxation of upper gradients or of the local Lipschitz constant of Lipschitz functions. The proof of this fact is inspired by a recent technique developed by Eriksson-Bique in Eriksson-Bique (2023 Calc. Var. Partial Differential Equations 62 23). In the possible non-complete setting, we prove that the chain Sobolev space is equal to the one defined via relaxation of the local Lipschitz constant of Lipschitz functions, while in general they are different from the one defined via upper gradients along curves. We apply the theory developed in the paper to prove equivalent formulations of the Poincaré inequality in terms of pointwise estimates involving ɛ-upper gradients, lower bounds on modulus of chains connecting points and size of separating sets measured with the Minkowski content in the non-complete setting. Along the way, we discuss the notion of weak ɛ-upper gradients and asymmetric notions of integral along chains.

In this paper, we study the Cauchy problem for pseudo-parabolic equations with a logarithmic nonlinearity. After establishing the existence and uniqueness of weak solutions within a suitable functional framework, we investigate several qualitative properties, including the asymptotic behaviour and blow-up of solutions as $t\to +\infty$. Moreover, when the initial data are close to a Gaussian function, we prove that these weak solutions exhibit either super-exponential growth or super-exponential decay.

Let $n\ge2$, $s\in(0,1)$, and $\Omega\subset\mathbb{R}^n$ be a bounded Lipschitz domain. In this paper, we investigate the global (higher-order) Sobolev regularity of weak solutions to the fractional Dirichlet problem

\begin{equation*}\begin{cases}(-\Delta)^su=f \ \ & \text{in}\ \ \Omega,\\u=0 \ \ & \text{in}\ \ \mathbb{R}^n\setminus\Omega.\end{cases}\end{equation*}

Precisely, we prove that there exists a positive constant $\varepsilon\in(0,s]$ depending on n, s, and the Lipschitz constant of Ω such that, for any $t\in[\varepsilon,\min\{1+\varepsilon,2s\})$, when $f\in L^q(\Omega)$ with some $q\in(\frac{n}{2s-t},\infty]$, the weak solution u satisfies

\begin{equation*}\|u\|_{W^{t,p}(\mathbb{R}^n)}\le C\|f\|_{L^q(\Omega)}\end{equation*}

for all $p\in[1,\frac{1}{t-\varepsilon})$. In particular, when Ω is a bounded C1 domain or a bounded Lipschitz domain satisfying the uniform exterior ball condition, the aforementioned global regularity estimates hold with $\varepsilon=s$ and they are sharp in this case. Moreover, if Ω is a bounded $C^{1,\kappa}$ domain with $\kappa\in(0,s)$ or a bounded Lipschitz domain satisfying the uniform exterior ball condition, we further show the global BMO-Sobolev regularity estimate

\begin{equation*}\left\|(-\Delta)^{\frac{s}{2}}u\right\|_{\mathrm{BMO}(\mathbb{R}^n)}+\left\|\nabla^{s}u\right\|_{\mathrm{BMO}(\mathbb{R}^n)}\le C\|f\|_{L^q(\Omega)}\end{equation*}

for some $q\in(\frac{n}{s},\infty]$, which is sharp in the sense that the BMO norm can not be improved to the $L^\infty$ norm.

We study the behaviour of the norm of the resolvent for non-self-adjoint operators of the form $A := -\partial_x + W(x)$, with $W(x) \ge 0$, defined in ${L^2}({\mathbb{R}})$. We provide a sharp estimate for the norm of its resolvent operator, $\| (A - \lambda)^{-1} \|$, as the spectral parameter diverges $(\lambda \to +\infty)$. Furthermore, we describe the C0-semigroup generated by −A and determine its norm. Finally, we discuss the applications of the results to the asymptotic description of pseudospectra of Schrödinger and damped wave operators, and also the optimality of abstract resolvent bounds based on Carleman-type estimates.

Factorization structures occur in toric differential and discrete geometry and can be viewed in multiple ways, e.g., as objects determining substantial classes of explicit toric Sasaki and Kähler geometries, as special coordinates on such or as an apex generalization of cyclic polytopes featuring a generalized Gale’s evenness condition. This article presents a comprehensive study of this new concept called factorization structures. It establishes their structure theory and introduces their use in the geometry of cones and polytopes. The article explains a construction of polytopes and cones compatible with a given factorization structure and exemplifies it for the product Segre–Veronese and Veronese factorization structures, where the latter case includes cyclic polytopes. Further, it derives the generalized Gale’s evenness condition for compatible cones, polytopes, and their duals and explicitly describes faces of these. Factorization structures naturally provide generalized Vandermonde identities, which relate normals of any compatible polytope, and which are used to find examples of Delzant and rational Delzant polytopes compatible with the Veronese factorization structure. The article offers a myriad of factorization structure examples, which are later characterized to be precisely factorization structures with decomposable curves, and raises the question if these encompass all factorization structures, i.e., the existence of an indecomposable factorization curve.

Numerous evolution equations with nonlocal convolution-type interactions have been proposed. In some cases, a convolution was imposed as the velocity in the advection term. Motivated by analysing these equations, we approximate advective nonlocal interactions as local ones, thereby converting the effect of nonlocality. In this study, we investigate whether the solution to the nonlocal Fokker–Planck equation can be approximated using the Keller–Segel system. By singular limit analysis, we show that this approximation is feasible for the Fokker–Planck equation with any potential and that the convergence rate is specified. Moreover, we provide an explicit formula for determining the coefficient of the Lagrange interpolation polynomial with Chebyshev nodes. Using this formula, the Keller–Segel system parameters for the approximation are explicitly specified by the shape of the potential in the Fokker–Planck equation. Consequently, we demonstrate the relationship between advective nonlocal interactions and a local dynamical system.

Here we consider the following fractional Hamiltonian system

\begin{equation*}\begin{cases}\begin{aligned}(-\Delta)^{s} u&=H_v(u,v) \;\;&&\text{in}~\Omega,\\(-\Delta)^{s} v&=H_u(u,v) &&\text{in}~\Omega,\\u &= v = 0 &&\text{in} ~ \mathbb{R}^N\setminus\Omega,\end{aligned}\end{cases}\end{equation*}

where $s\in (0,1)$, $N \gt 2s$, $H \in C^1(\mathbb{R}^2, \mathbb{R})$, and $\Omega \subset \mathbb{R}^N$ is a smooth bounded domain. To apply the variational method for this problem, the key question is to find a suitable functional setting. Instead of usual fractional Sobolev spaces, we use the solution space of $(-\Delta)^{s}u=f\in L^r(\Omega)$ for $r\ge 1$, for which we show the (compact) embedding properties. When H has subcritical and superlinear growth, we construct two frameworks, respectively with the interpolation space method and the dual method, to show the existence of nontrivial solution. As byproduct, we revisit the fractional Lane–Emden system, i.e. $H(u, v)=\frac{1}{p+1}|u|^{p+1}+\frac{1}{q+1}|v|^{q+1}$, and consider the existence, uniqueness of (radial) positive solutions under subcritical assumption.

This paper is the latter part of a series of our studies on the concentration and oscillation analysis of semilinear elliptic equations with exponential growth $e^{u^p}$. In the first one [17], we completed the concentration analysis of blow-up positive solutions in the supercritical case p > 2 via a scaling approach. As a result, we detected infinite sequences of concentrating parts with precise quantification. In the present paper, we proceed to our second aim, the oscillation analysis. Especially, we deduce an infinite oscillation estimate directly from the previous infinite concentration ones. This allows us to investigate intersection properties between blow-up solutions and singular functions. Consequently, we show that the intersection number between blow-up and singular solutions diverges to infinity. This leads to a proof of infinite oscillations of bifurcation diagrams, which ensures the existence of infinitely many solutions. Finally, we also remark on infinite concentration and oscillation phenomena in the limit cases $p\to2^+$ and $p\to \infty$.

We prove the convergence of a Wasserstein gradient flow of a free energy in inhomogeneous media. Both the energy and media can depend on the spatial variable in a fast oscillatory manner. In particular, we show that the gradient-flow structure is preserved in the limit, which is expressed in terms of an effective energy and Wasserstein metric. The gradient flow and its limiting behavior are analysed through an energy dissipation inequality. The result is consistent with asymptotic analysis in the realm of homogenisation. However, we note that the effective metric is in general different from that obtained from the Gromov–Hausdorff convergence of metric spaces. We apply our framework to a linear Fokker–Planck equation, but we believe the approach is robust enough to be applicable in a broader context.

We introduce a notion of equivariant vector bundles on schemes over semirings. We do this by considering the functor of points of a locally free sheaf. We prove that every toric vector bundle on a toric scheme X over an idempotent semifield equivariantly splits as a sum of toric line bundles. We then study the equivariant Picard group $\operatorname{Pic}_G(X)$. Finally, we prove a version of Klyachko’s classification theorem for toric vector bundles over an idempotent semifield.

For an even positive integer n, we study rank-one Eisenstein cohomology of the split orthogonal group $\mathrm {O}(2n+2)$ over a totally real number field $F.$ This is used to prove a rationality result for the ratios of successive critical values of degree-$2n$ Langlands L-functions associated to the group $\mathrm {GL}_1 \times \mathrm {O}(2n)$ over F. The case $n=2$ specializes to classical results of Shimura on the special values of Rankin–Selberg L-functions attached to a pair of Hilbert modular forms.

We focus on the existence of ground state solutions to the following class of elliptic equations

\begin{equation*}-\Delta_{\mathbb{B}^N} u+V(x)u=f(u) \quad \hbox{in} \quad \mathbb{B}^N,\end{equation*}

where $\mathbb{B}^N$ is the disc model of the Hyperbolic space and $\Delta_{\mathbb{B}^N}$ denotes the Laplace–Beltrami operator with $N \geq 2$, $V:\mathbb{B}^N \to \mathbb{R}$ and $f:\mathbb{R} \to \mathbb{R}$ are continuous functions that satisfy some technical conditions. With different types of the potential V, by introducing some new tricks handling the hurdle that the Hyperbolic space is not a compact manifold, we are able to obtain at least a positive ground state solution using variational methods.

As some applications for the methods adopted above, we derive the existence of normalized solutions to the elliptic problems

\begin{equation*}\left\{\begin{aligned}&-\Delta_{\mathbb{B}^N} u+V(x)u=\mu u+f(u) \quad \hbox{in} \quad \mathbb{B}^N,\\&\int_{\mathbb{B}^N}|u|^{2}\,dV_{{\mathbb{B}^N}}=a^{2},\end{aligned}\right.\end{equation*}

where a > 0, $\mu\in \mathbb{R}$ is an unknown parameter that appears as a Lagrange multiplier and f is a continuous function that fulfils the L2-subcritical or L2-supercritical growth. We do believe that it seems the first results to deal with normalized solutions for the Schrödinger equations in the Hyperbolic space.

We investigate radial and non-radial solutions to a class of (p, q)-Laplace equations involving weights. More precisely, we obtain existence and multiplicity results for nontrivial nonnegative radial and non-radial solutions, which extend results in the literature. Moreover, we study the non-radiality of minimizers in Hénon type (p, q)-Laplace problems and symmetry-breaking phenomena.

Dedicated to Professor Pavel Drábek on the occasion of his seventieth birthday

A family of arbitrarily high-order energy-preserving methods are developed to solve the coupled Schrödinger–Boussinesq (S-B) system. The system is a nonlinear coupled system and satisfies a series of conservation laws. It is often difficult to construct a high-order decoupling numerical algorithm to solve the nonlinear system. In this paper, the original system is first reformulated into an equivalent Hamiltonian system by introducing multiple auxiliary variables. Next, the reformulated system is discretized by the Fourier pseudo-spectral method and the implicit midpoint scheme in the spatial and temporal directions, respectively, and a second-order conservative scheme is obtained. Finally, the scheme is extended to arbitrarily high-order accuracy by means of diagonally implicit symplectic Runge–Kutta methods or composition methods. Rigorous analyses show that the proposed methods are fully decoupled and can precisely conserve the discrete invariants. Numerical results show that the proposed schemes are effective and can be easily extended to other nonlinear partial differential equations.

We establish the higher fractional differentiability for the minimizers of non-autonomous integral functionals of the form

\begin{align*}\mathcal{F}(u,\Omega):=\int_\Omega \left[ f(x,Du)- g \cdot u \right] dx ,\end{align*}

under (p, q)-growth conditions. Besides a suitable differentiability assumption on the partial map $x \mapsto D_\xi f(x,\xi)$, we do not need to assume any differentiability assumption on the function g.

We investigate the Lorentzian analogues of Riemannian Bianchi–Cartan–Vranceanu spaces. We provide their general description and emphasize their role in the classification of three-dimensional homogeneous Lorentzian manifolds with a four-dimensional isometry group. We then illustrate their geometric properties (with particular regard to curvature, Killing vector fields and their description as Lorentzian Lie groups) and we study several relevant classes of surfaces (parallel, totally umbilical, minimal, constant mean curvature) in these homogeneous Lorentzian three-manifolds.

We investigate the consequences of periodic, on–off glucose infusion on the glucose–insulin regulatory system based on a system-level mathematical model with two explicit time delays. Studying the effects of such infusion protocols is mathematically challenging yet a promising direction for probing the system response to infusion. We pay special attention to the interplay of periodic infusion with intermediate-time-scale, ultradian oscillations that arise as a result of the physiological response of glucose uptake and back-release into the bloodstream. By using numerical solvers and numerical continuation software, we investigate the response of the model to different infusion patterns, explore how these patterns affect the overall levels of glucose and insulin, and how this can lead to entrainment. By doing so, we provide a road-map of system responses that can potentially help identify new, less-invasive, test strategies for detecting abnormal responses to glucose uptake without falling into lockstep with the infusion pattern.