Macroscopically, a Darcian unsaturated moisture flow in the top soil is usually represented by an one-dimensional volume scale of evaporation from a static water table. On the microscale, simple pore-level models posit bundles of small-radius capillary tubes of a constant circular cross-section, fully occupied by mobile water moving in the Hagen–Poiseuille (HP) regime, while large-diameter pores are occupied by stagnant air. In our paper, cross-sections of cylindrical pores are polygonal. Steady, laminar, fully developed two-dimensional flows of Newtonian water in prismatic conduits, driven by a constant pressure gradient along a pore gradient, are more complex than the HP formula; this is based on the fact that the pores are only partially occupied by water and immobile air. The Poisson equation in a circular tetragon, with no-slip or mixed (no-shear-stress) boundary conditions on the two adjacent pore walls and two menisci, is solved by the methods of complex analysis. The velocity distribution is obtained via the Keldysh–Sedov type of singular integrals, and the flow rate is evaluated for several sets of meniscus radii by integrating the velocity over the corresponding tetragons.

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.

An area-preserving homeomorphism isotopic to the identity is said to have rational rotation direction if its rotation vector is a real multiple of a rational class. We give a short proof that any area-preserving homeomorphism of a compact surface of genus at least two, which is isotopic to the identity and has rational rotation direction, is either the identity or has periodic points of unbounded minimal period. This answers a question of Ginzburg and Seyfaddini and can be regarded as a Conley conjecture-type result for symplectic homeomorphisms of surfaces beyond the Hamiltonian case. We also discuss several variations, such as maps preserving arbitrary Borel probability measures with full support, maps that are not isotopic to the identity and maps on lower genus surfaces. The proofs of the main results combine topological arguments with periodic Floer homology.

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.

Using a recent result of Bowden, Hensel and Webb, we prove the existence of a homeomorphism with positive stable commutator length in the group of homeomorphisms of the Klein bottle which are isotopic to the identity.

In this paper, we prove the non-existence of Codazzi and totally umbilical hypersurfaces, especially totally geodesic hypersurfaces, in the $4$-dimensional model space $\mathrm {Sol}_1^4$.

We consider a pair of identical theta neurons in the active regime, each coupled to the other via a delayed Dirac delta function. The network can support periodic solutions and we concentrate on solutions for which the neurons are half a period out of phase with one another, and also solutions for which the neurons are perfectly synchronous. The dynamics are analytically solvable, so we can derive explicit expressions for the existence and stability of both types of solutions. We find two branches of solutions, connected by symmetry-broken solutions which arise when the period of a solution as a function of delay is at a maximum or a minimum.

We study the descriptive complexity of sets of points defined by restricting the statistical behaviour of their orbits in dynamical systems on Polish spaces. Particular examples of such sets are the sets of generic points of invariant Borel probability measures, but we also consider much more general sets (for example, $\alpha $-Birkhoff regular sets and the irregular set appearing in the multifractal analysis of ergodic averages of a continuous real-valued function). We show that many of these sets are Borel in general, and all these are Borel when we assume that our space is compact. We provide examples of these sets being non-Borel, properly placed at the first level of the projective hierarchy (they are complete analytic or co-analytic). This proves that the compactness assumption is, in some cases, necessary to obtain Borelness. When these sets are Borel, we measure their descriptive complexity using the Borel hierarchy. We show that the sets of interest are located at most at the third level of the hierarchy. We also use a modified version of the specification property to show that these sets are properly located at the third level of the hierarchy for many dynamical systems. To demonstrate that the specification property is a sufficient, but not necessary, condition for maximal descriptive complexity of a set of generic points, we provide an example of a compact minimal system with an invariant measure whose set of generic points is $\boldsymbol {\Pi }^0_3$-complete.

In this paper, we derive simple analytical bounds for solutions of $x - \ln x = y -\ln y$, and use them for estimating trajectories following Lotka–Volterra-type integrals. We show how our results give estimates for the Lambert W function as well as for trajectories of general predator–prey systems, including, for example, Rosenzweig–MacArthur equations.

Machine learning has exhibited substantial success in the field of natural language processing (NLP). For example, large language models have empirically proven to be capable of producing text of high complexity and cohesion. However, at the same time, they are prone to inaccuracies and hallucinations. As these systems are increasingly integrated into real-world applications, ensuring their safety and reliability becomes a primary concern. There are safety critical contexts where such models must be robust to variability or attack and give guarantees over their output. Computer vision had pioneered the use of formal verification of neural networks for such scenarios and developed common verification standards and pipelines, leveraging precise formal reasoning about geometric properties of data manifolds. In contrast, NLP verification methods have only recently appeared in the literature. While presenting sophisticated algorithms in their own right, these papers have not yet crystallised into a common methodology. They are often light on the pragmatical issues of NLP verification, and the area remains fragmented. In this paper, we attempt to distil and evaluate general components of an NLP verification pipeline that emerges from the progress in the field to date. Our contributions are twofold. First, we propose a general methodology to analyse the effect of the embedding gap – a problem that refers to the discrepancy between verification of geometric subspaces, and the semantic meaning of sentences which the geometric subspaces are supposed to represent. We propose a number of practical NLP methods that can help to quantify the effects of the embedding gap. Second, we give a general method for training and verification of neural networks that leverages a more precise geometric estimation of semantic similarity of sentences in the embedding space and helps to overcome the effects of the embedding gap in practice.

In this paper, we establish Newton–Maclaurin-type inequalities for functions arising from linear combinations of primitively symmetric polynomials. This generalization extends the classical Newton–Maclaurin inequality to a broader class of functions.

We consider the Cauchy problem of the non-linear Schrödinger equation with the modulated dispersion and power type non-linearities in any spatial dimensions. We adapt the Young integral theory developed by Chouk–Gubinelli [7] and multilinear estimates which are based on divisor counting and show the local well-posedness. This generalizes the result by Chouk–Gubinelli [7] in terms of the dimension and the order of the non-linearity.

This work investigates the online machine learning problem of prediction with expert advice in an adversarial setting through numerical analysis of, and experiments with, a related partial differential equation. The problem is a repeated two-person game involving decision-making at each step informed by $n$ experts in an adversarial environment. The continuum limit of this game over a large number of steps is a degenerate elliptic equation whose solution encodes the optimal strategies for both players. We develop numerical methods for approximating the solution of this equation in relatively high dimensions ($n\leq 10$) by exploiting symmetries in the equation and the solution to drastically reduce the size of the computational domain. Based on our numerical results we make a number of conjectures about the optimality of various adversarial strategies, in particular about the non-optimality of the COMB strategy.

This study explores the dynamics of a simple mechanical oscillator involving a magnet on a spring constrained to an axis; this magnet is additionally subject to the attractive force from a second magnet, which is placed on a parallel offset axis. The moments of both magnets remain aligned. The dynamics of the first magnet is first analysed in isolation for an unforced situation in which the second magnet is static and its position is taken as a parameter. We find codimension-1 saddle-node bifurcations, as well as a codimension-2 cusp bifurcation. The system has a region of bistability which increases in size with increasing force ratio. Next, the parametrically forced situation is considered, in which the second magnet moves sinusoidally. A comprehensive analysis of the forced oscillator behaviour is presented from the dynamical-systems standpoint. The solutions are shown to include periodic, quasiperiodic and chaotic trajectories. Resonances are shown to exist and the effect of weak damping is explored. Layered stroboscopic maps are used to produce cross-sections of the chaotic attractor as the parametric forcing frequency is varied. The strange attractor is found to disappear for a narrow window of forcing frequencies near the natural frequency of the spring.

In this article, we investigate the following non-linear Schrödinger (NLS) equation with Neumann boundary conditions:

\begin{equation*}\begin{cases}-\Delta u+ \lambda u= f(u) & {\mathrm{in}} \,~ \Omega,\\\displaystyle\frac{\partial u}{\partial \nu}=0 \, &{\mathrm{on}}\,~\partial \Omega\end{cases}\end{equation*}

coupled with a constraint condition:

\begin{equation*}\int_{\Omega}|u|^2 dx=c,\end{equation*}

where $\Omega\subset \mathbb{R}^N(N\ge3)$ denotes a smooth bounded domain, ν represents the unit outer normal vector to $\partial \Omega$, c is a positive constant, and λ acts as a Lagrange multiplier. When the non-linearity f exhibits a general mass supercritical growth at infinity, we establish the existence of normalized solutions, which are not necessarily positive solutions and can be characterized as mountain pass type critical points of the associated constraint functional. Our approach provides a uniform treatment of various non-linearities, including cases such as $f(u)=|u|^{p-2}u$, $|u|^{q-2}u+ |u|^{p-2}u$, and $-|u|^{q-2}u+|u|^{p-2}u$, where $2 \lt q \lt 2+\frac{4}{N} \lt p \lt 2^*$. The result is obtained through a combination of a minimax principle with Morse index information for constrained functionals and a novel blow-up analysis for the NLS equation under Neumann boundary conditions.

While constructing mathematical models, scientists usually consider biotic factors, but it is crystal-clear that abiotic factors, such as wind, are also important as biotic factors. From this point of view, this paper is devoted to the investigation of some bifurcation properties of a fractional-order prey–predator model under the effect of wind. Using fractional calculus is very popular in modelling, since it is more effective than classical calculus in predicting the system’s future state and also discretization is one of the most powerful tools to study the behaviour of the models. In this paper, first of all, the model is discretized by using a piecewise discretization approach. Then, the local stability of fixed points is considered. We show using the centre manifold theorem and bifurcation theory that the system experiences a flip bifurcation and a Neimark–Sacker bifurcation at a positive fixed point. Finally, numerical simulations are given to demonstrate our results.

This article is concerned with the spreading speed and traveling waves of a lattice prey–predator system with non-local diffusion in a periodic habitat. With the help of an associated scalar lattice equation, we derive the invasion speed for the predator. More specifically, when the dispersal kernel of the predator is exponentially bounded, the invasion speed is finite and can be characterized in terms of principal eigenvalues; while the dispersal kernel is algebraically decaying, the invasion speed is infinite and the accelerated spreading rate is obtained. Furthermore, the existence and non-existence of traveling waves connecting the semi-equilibrium point to a uniformly persistent state are established.