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.

Consider, for any integer $n\ge 3$, the set $\operatorname {\mathrm {Pos}}_n$ of all n-periodic tree patterns with positive topological entropy and the set $\operatorname {\mathrm {Irr}}_n\subset \operatorname {\mathrm {Pos}}_n$ of all n-periodic irreducible tree patterns. The aim of this paper is to determine the elements of minimum entropy in the families $\operatorname {\mathrm {Pos}}_n$, $\operatorname {\mathrm {Irr}}_n$ and $\operatorname {\mathrm {Pos}}_n\setminus \operatorname {\mathrm {Irr}}_n$. Let $\unicode{x3bb} _n$ be the unique real root of the polynomial $x^n-2x-1$ in $(1,+\infty )$. We explicitly construct an irreducible n-periodic tree pattern $\mathcal {Q}_n$ whose entropy is $\log (\unicode{x3bb} _n)$. We prove that this entropy is minimum in $\operatorname {\mathrm {Pos}}_n$. Since the pattern $\mathcal {Q}_n$ is irreducible, $\mathcal {Q}_n$ also minimizes the entropy in the family $\operatorname {\mathrm {Irr}}_n$. We also prove that the minimum positive entropy in the set $\operatorname {\mathrm {Pos}}_n\setminus \operatorname {\mathrm {Irr}}_n$ (which is non-empty only for composite integers $n\ge 6$) is $\log (\unicode{x3bb} _{n/p})/p$, where p is the least prime factor of n.

We prove that every locally compact second countable group G arises as the outer automorphism group $\operatorname{Out} M$ of a II1 factor, which was so far only known for totally disconnected groups, compact groups, and a few isolated examples. We obtain this result by proving that every locally compact second countable group is a centralizer group, a class of Polish groups that arise naturally in ergodic theory and that may all be realized as $\operatorname{Out} M$.

Deep neural networks and other modern machine learning models are often susceptible to adversarial attacks. Indeed, an adversary may often be able to change a model’s prediction through a small, directed perturbation of the model’s input – an issue in safety-critical applications. Adversarially robust machine learning is usually based on a minmax optimisation problem that minimises the machine learning loss under maximisation-based adversarial attacks. In this work, we study adversaries that determine their attack using a Bayesian statistical approach rather than maximisation. The resulting Bayesian adversarial robustness problem is a relaxation of the usual minmax problem. To solve this problem, we propose Abram – a continuous-time particle system that shall approximate the gradient flow corresponding to the underlying learning problem. We show that Abram approximates a McKean–Vlasov process and justify the use of Abram by giving assumptions under which the McKean–Vlasov process finds the minimiser of the Bayesian adversarial robustness problem. We discuss two ways to discretise Abram and show its suitability in benchmark adversarial deep learning experiments.

We provide a complete classification of Teichmüller curves occurring in hyperelliptic components of the meromorphic strata of differentials. Using a non-existence criterion based on how Teichmüller curves intersect the boundary of the moduli space we derive a contradiction to the algebraicity of any candidate outside of Hurwitz covers of strata with projective dimension one, and Hurwitz covers of zero residue loci in strata with projective dimension two.

This paper studies twisted signature invariants and twisted linking forms, with a view toward obstructions to knot concordance. Given a knot K and a representation $\rho $ of the knot group, we define a twisted signature function $\sigma _{K,\rho } \colon S^1 \to \mathbb {Z}$. This invariant satisfies many of the same algebraic properties as the classical Levine-Tristram signature $\sigma _K$. When the representation is abelian, $\sigma _{K,\rho }$ recovers $\sigma _K$, while for appropriate metabelian representations, $\sigma _{K,\rho }$ is closely related to the Casson-Gordon invariants. Additionally, we prove satellite formulas for $\sigma _{K,\rho }$ and for twisted Blanchfield forms.

Given a general polarized $K3$ surface $S\subset \mathbb P^g$ of genus $g\le 14$, we study projections of minimal degree and their variational structure. In particular, we prove that the degree of irrationality of all such surfaces is at most $4$, and that for $g=7,8,9,11$ there are no rational maps of degree $3$ induced by the primitive linear system. Our methods combine vector bundle techniques à la Lazarsfeld with derived category tools and also make use of the rich theory of singular curves on $K3$ surfaces.

We conduct a theoretical analysis of the performance of $\beta $-encoders. The $\beta $-encoders are A/D (analogue-to-digital) encoders, the design of which is based on the expansion of real numbers with noninteger radix. For the practical use of such encoders, it is important to have theoretical upper bounds of their errors. We investigate the generating function of the Perron–Frobenius operator of the corresponding one-dimensional map and deduce the invariant measure of it. Using this, we derive an approximate value of the upper bound of the mean squared error of the quantization process of such encoders. We also discuss the results from a numerical viewpoint.

We study the behaviour of Kauffman bracket skein modules of 3-manifolds under gluing along surfaces. For this we extend this notion to $3$-manifolds with marking consisting of open intervals and circles in the boundary. The new module is called the stated skein module.

The first results concern non-injectivity of certain natural maps defined when forming connected sums along spheres or disks. These maps are injective for surfaces or for generic quantum parameter, but we show that in general they are not when the quantum parameter is a root of 1. We show that when the quantum parameter is a root of 1, the empty skein is zero in a connected sum where each constituent manifold has non-empty marking. We also prove various non-injectivity results for the Chebyshev-Frobenius map and the map induced by deleting marked balls.

We then interpret stated skein modules as a monoidal symmetric functor from a category of “decorated cobordisms” to a category of algebras and their bimodules. We apply this to deduce properties of stated skein modules as a Van-Kampen like theorem, a computation through Heegaard decompositions and a relation to Hochshild homology for trivial circle bundles over surfaces.