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.

It is shown that if $\{H_n\}_{n \in \omega}$ is a sequence of groups without involutions, with $1 \lt |H_n| \leq 2^{\aleph_0}$, then the topologist’s product modulo the finite words is (up to isomorphism) independent of the choice of sequence. This contrasts with the abelian setting: if $\{A_n\}_{n \in \omega}$ is a sequence of countably infinite torsion-free abelian groups, then the isomorphism class of the product modulo sum $\prod_{n \in \omega} A_n/\bigoplus_{n \in \omega} A_n$ is dependent on the sequence.

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.

In this article, we study the effect of the Hardy potential on existence, uniqueness, and optimal summability of solutions of the mixed local–nonlocal elliptic problem

\begin{equation*}-\Delta u + (-\Delta)^s u - \gamma \frac{u}{|x|^2}=f \,\text{in } \Omega, \ u=0 \,\text{in } {\mathbb R}^n \setminus \Omega,\end{equation*}

${\mathbb R}^n$

Let S be an orientable, connected surface of finite topological type, with genus $g \leqslant 2$, empty boundary and complexity at least 2; we prove that any graph endomorphism of the curve graph of S is actually an automorphism. Also, as a complement of the author’s previous results, we prove that under mild conditions on the complexity of the underlying surfaces, any graph morphism between curve graphs is induced by a homeomorphism of the surfaces.

To prove these results, we construct a finite subgraph whose union of iterated rigid expansions is the curve graph $\mathcal{C}(S)$. The sets constructed, and the method of rigid expansion, are closely related to Aramayona and Leininger’s finite rigid sets. We prove as a consequence that Aramayona and Leininger’s rigid set also exhausts $\mathcal{C}(S)$ via rigid expansions. The combinatorial rigidity results follow as an immediate consequence, based on the author’s previous results.

We show that the loop space of a moment-angle complex associated to a two-dimensional simplicial complex decomposes as a finite type product of spheres, loops on spheres and certain indecomposable spaces which appear in the loop space decomposition of Moore spaces. We also give conditions on certain subcomplexes under which, localised away from sufficiently many primes, the loop space of a moment-angle complex decomposes as a finite type product of spheres and loops on spheres.

Let (W, S) be a Coxeter system of rank n, and let $p_{(W, S)}(t)$ be its growth function. It is known that $p_{(W, S)}(q^{-1}) \lt \infty$ holds for all $n \leq q \in \mathbb{N}$. In this paper, we will show that this still holds for $q = n-1$, if (W, S) is 2-spherical. Moreover, we will prove that $p_{(W, S)}(q^{-1}) = \infty$ holds for $q = n-2$, if the Coxeter diagram of (W, S) is the complete graph. These two results provide a complete characterization of the finiteness of the growth function in the case of 2-spherical Coxeter systems with a complete Coxeter diagram.

Chan and Seceleanu have shown that if a weighted shift operator on $\ell^p(\mathbb{Z})$, $1\leq p \lt \infty$, admits an orbit with a non-zero limit point then it is hypercyclic. We present a new proof of this result that allows to extend it to very general sequence spaces. In a similar vein, we show that, in many sequence spaces, a weighted shift with a non-zero weakly sequentially recurrent vector has a dense set of such vectors, but an example on $c_0(\mathbb{Z})$ shows that such an operator is not necessarily hypercyclic. On the other hand, we obtain that weakly sequentially hypercyclic weighted shifts are hypercyclic. Chan and Seceleanu have, moreover, shown that if an adjoint multiplication operator on a Bergman space admits an orbit with a non-zero limit point then it is hypercyclic. We extend this result to very general spaces of analytic functions, including the Hardy spaces.

We study quotients of mapping class groups of punctured spheres by suitable large powers of Dehn twists, showing an analogue of Ivanov’s theorem for the automorphisms of the corresponding quotients of curve graphs. Then we use this result to prove quasi-isometric rigidity of these quotients, answering a question of Behrstock, Hagen, Martin, and Sisto in the case of punctured spheres. Finally, we show that the automorphism groups of our quotients of mapping class groups are “small”, as are their abstract commensurators. This is again an analogue of a theorem of Ivanov about the automorphism group of the mapping class group.

In the process, we develop techniques to extract combinatorial data from a quasi-isometry of a hierarchically hyperbolic space, and use them to give a different proof of a result of Bowditch about quasi-isometric rigidity of pants graphs of punctured spheres.

Minimizing an adversarial surrogate risk is a common technique for learning robust classifiers. Prior work showed that convex surrogate losses are not statistically consistent in the adversarial context – or in other words, a minimizing sequence of the adversarial surrogate risk will not necessarily minimize the adversarial classification error. We connect the consistency of adversarial surrogate losses to properties of minimizers to the adversarial classification risk, known as adversarial Bayes classifiers. Specifically, under reasonable distributional assumptions, a convex surrogate loss is statistically consistent for adversarial learning iff the adversarial Bayes classifier satisfies a certain notion of uniqueness.

We show that the group $ \langle a,b,c,t \,:\, a^t=b,b^t=c,c^t=ca^{-1} \rangle$ is profinitely rigid amongst free-by-cyclic groups, providing the first example of a hyperbolic free-by-cyclic group with this property.