Papers

Despite the popularity of Koopman modeling for nonlinear systems, in the presence of input variables, the evident nonexistence of a fully linear time-invariant model even in infinite dimensions makes Koopman-based control largely an open problem to date. Focusing on discrete-time systems in this paper, which eschews from using operator semigroup and infinitesimal generator notions, it is proven that nonlinear systems, if satisfying appropriate smoothness and regularity conditions, can be expressed exactly as bilinear dynamics, when the state variables and input variables are separately lifted into their reproducing kernel Hilbert spaces (RKHSs). To account for the knowledge of an equilibrium point at the origin, the RKHS is defined by a linear–radial product kernel, and hence the functions belonging to this RKHS are spanned by the multiplications of component functions and Sobolev functions. The stabilization problem, namely the determination of a feedback law that causes a Lyapunov function (expressed as a kernel sum-of-squares form) to decrease, is then posed as an infinite-dimensional optimization problem over state-dependent conditional probability measures over the input space, solved via a discretization scheme.

We apply a deep-neural-network (DNN) ansatz to the symmetrized spectator vector of the subtracted three-body Faddeev integral equation for identical bosons near the Efimov limit. The network is trained by minimizing the residual of the discretized integral equation, while the positive binding scale associated with the three-body energy is treated as a trainable parameter. Deterministic diagonalization of the same discretized kernel is used only as an a posteriori numerical benchmark. As preliminary validation, the neural-solver strategy is tested on the analytically solvable hydrogen radial problem. At unitarity, the DNN reproduces the Efimov ground-state binding scale with a DNN–deterministic deviation of $0.022\%$, while the first excited state is recovered to $0.002\%$. The deterministic solver recovers the universal Efimov scaling ratio $e^{2\pi/s_0}\simeq 515.03$, and the neural method traces the bound-state branches as a function of the inverse scattering length $1/a$ by continuation from the unitary solution. These results indicate that DNN-based residual minimization can provide a compact and differentiable representation of a renormalized few-body integral-equation solution in a regime governed by discrete scale invariance.

Building on preliminary results about lower sections of real-valued functions on a metric space, we provide a constructive counterpart of Ekeland's theorem on the approximate optimisation of real-valued functions.

We give a novel proof of the Gorin-Shkolnikov identity based on Pitman's SDE representation of Brownian excursion local times. More generally, we derive a family of Gaussian identities for nonlinear functionals of excursion local times, which includes Hariya's identity. These results extend naturally to reflected Brownian bridges conditioned on their local times and to Brownian meanders.

Continuous approximations of flow data are useful for downstream analysis, differentiation, and visualization, but purely data-driven reconstructions do not, in general, preserve the governing physics. This limitation becomes particularly important when input data are physically inconsistent, whether due to low-fidelity discretizations or unmodeled discrepancies. In such cases, reconstructed fields may exhibit inaccurate PDE residuals, violated balance laws, or unreliable derived quantities. To address this, we propose a physics-informed B-spline framework that embeds physical constraints directly into the reconstruction process. The method constructs compact, continuously differentiable representations of discrete fields using tensor-product B-splines and determines spline control points by solving an optimization problem balancing data fidelity with residuals of the governing PDEs, alongside initial and boundary conditions. Leveraging exact analytical derivatives of the B-spline basis enables efficient and accurate evaluation of physical residuals without storing full-resolution fields. We refer to this approach as physics-informed multivariate functional approximation (PI-MFA). Numerical studies on the 1D convection-diffusion, 2D coupled Burgers, and 2D incompressible Navier-Stokes equations show PI-MFA reduces PDE residuals and improves global balance-law consistency. Compared with standard and regularized MFA, PI-MFA produces more physically faithful reconstructions and, for physically inconsistent data, lower approximation errors, while offering computational advantages over tested physics-informed neural networks. Overall, PI-MFA preserves the compactness, local support, and exact differentiability of classical spline spaces while producing reliable continuous flow fields for scientific analysis and visualization.

We provide progress on the characterization of composite integers $n$ that satisfy the jump conditions $\varphi(n+12)=\varphi(n)+12$ and $\sigma(n+12)=\sigma(n)+12$ simultaneously. While it is known that prime quadruplets $(p,p+2,p+6,p+8)$ generate solutions $n=p(p+8)$, the complete characterization remains an open conjecture. We prove that this characterization is complete when $n$ and $n+12$ are both squarefree semiprimes, and that no solution $n$ can be a prime power. Furthermore, a complete search up to $10^{12}$ resulted in no counterexamples to the conjecture. If this conjecture is proven true, and there are infinitely many such solutions, then it can be proved that there are infinitely many prime quadruplets.

In 2025 Cornet, Dravec, and Torres determined the domination number $\gamma(J(n,3))$ of the Johnson graph for every even $n \ge 6$, expressing it as a closed form $\phi_n$ in terms of Fort-Hedlund covering numbers, and conjectured the same value for odd $n$. We prove this conjecture: $\gamma(J(n,3)) = \phi_n$ for every odd $n \ge 7$. A dominating set of $J(n,3)$ is the same as a lotto design $\mathrm{LD}(n,3,3,2)$, and the lotto numbers $L(n,3,3,2)$, and hence $\gamma(J(n,3))$, were already determined for every $n$ by Bate (1978) and by Brouwer (1981); a short degree argument of Füredi, Székely, and Zubor (1996) gives the same lower bound for all $n \not\equiv 9, 10 \pmod{12}$. The contribution of this paper is therefore a new proof of the odd case, carried out within the bijective framework of Cornet, Dravec, and Torres.

Neural combinatorial optimization (NCO) trains autoregressive policies to solve routing problems. The standard training algorithm, REINFORCE with a rollout baseline, requires maintaining and periodically updating a frozen copy of the policy for variance reduction. This baseline introduces a structural vulnerability: on harder instances, a poor baseline produces noisy gradient estimates that can destabilize training. We evaluate Group Relative Policy Optimization (GRPO), an algorithm from large language model alignment that eliminates the baseline entirely by normalizing advantages within groups of sampled trajectories. In a controlled comparison of five RL algorithms on TSP and CVRP benchmarks within the RL4CO framework, we find that: (i) GRPO avoids the training collapse observed with REINFORCE on TSP-100, where performance degrades from cost 9.8 to 52.1 immediately after the warmup phase and does not recover under extended training; (ii) at matched gradient updates, GRPO achieves solution quality within 2% of POMO, a strong AM-based multi-start baseline, while requiring no external baseline; and (iii) P3O, a pairwise preference algorithm also from the alignment literature, is competitive on TSP but shows higher variability on CVRP. These results identify GRPO as a promising baseline-free alternative for NCO, particularly in settings where baseline-dependent training becomes fragile.

In this paper, we prove an invariance principle for the additive functionals of a family of voter models, which include the nearest-neighbor cases on $d$-dimensional lattices for $d$ at least $5$ or on regular trees with degree at least $3$ and some long-range cases on lattices as special examples. The proof of our main result extends a martingale decomposition method, where the Kolmogorov backward equation and the duality relationship between the voter model and the coalescing random walk play the key roles.

We present several Laplacian-type matrices associated with a loopless digraph $D$: the out-/in-degree Laplacians $\mathcal L_{\mathrm{out}},\mathcal L_{\mathrm{in}}$, the incidence Laplacian $\mathcal L_{\mathrm{inc}}=BB^{\mathsf T}$, and the symmetrized and skew-symmetrized variants $\mathcal S_{\mathrm{out}},\mathcal K_{\mathrm{out}}$. We show that $\mathcal L_{\mathrm{inc}}(D)$ coincides with the Laplacian of the underlying undirected multigraph, and we derive spectral and characteristic-polynomial relations under arc reversal and complementation (including a simplification for Eulerian digraphs for $\mathcal S_{\mathrm{out}}$). We demonstrate that the spectral radius of $\mathcal L_{\mathrm{out}}$ is bounded above by the order of the digraph and give a characterization in the equality case. We further obtain explicit formulas for joins and line digraphs, giving a general determinantal identity relating the out-degree Laplacian characteristic polynomials of a regular digraph and its line digraph.

The Gromov–Wasserstein (GW) distance provides a framework for comparing metric measure spaces, regardless of their underlying structure or geometry. For network-based data, it enables direct comparisons of graphs with different numbers of nodes, without requiring an embedding or other abstraction. Furthermore, through a variant of GW known as fused Gromov–Wasserstein (fGW), it is also possible to incorporate node features in addition to graph structure. In this work, we implement $k$-nearest neighbors ($k$-NN) classification using the GW and fGW distances. We prove the universal consistency of the GW-$k$-NN classifier on the space of equivalence classes of metric measure spaces with finite support and uniform probability measure. By viewing graphs as finitely supported metric measure spaces equipped with the pairwise distance metric and a uniform probability measure on the nodes, we obtain universal consistency of GW-$k$-NN for the space of graphs. Likewise for fGW-$k$-NN, we prove universal consistency on the space of weak isomorphism classes of structured objects consisting of metric measure spaces with finite support and uniform probability measure and feature maps into Euclidean space, thus establishing universal consistency on the space of node-attributed graphs. Our numerical experiments show that GW-$k$-NN and fGW-$k$-NN consistently perform well across multiple graph datasets, suggesting that metric classifiers such as $k$-NN work well in the GW framework.

Several recent papers have demonstrated the utility of using Lie groups within estimation problems, yielding improved accuracy and consistency. This paper introduces a new tool for describing operations with matrix Lie groups: tensors and the Einstein summation notation. While tensors and Einstein notation are well-known in other research fields, applying this mathematical notation to represent and compute matrix Lie derivatives is novel. More importantly, this new notation greatly clarifies the derivatives and operations necessary to work with matrix Lie Groups in (gradient-based) estimation frameworks. Therefore, the main contribution of this paper is not a new capability, but a more perspicuous mathematical notation for working with matrix Lie groups.

For nearly a century, the transition state—the structure organizing motion between reactants and products—has seemed to lack an exact quantum counterpart. Its geometry—a recrossing-free dividing surface whose one-way flux gives the exact rate—seems to require position and momentum direction at once, in tension with the uncertainty principle. The obstruction seems fundamental. It is not. The exact quantum equations generate a phase-space flow whose stable and unstable manifolds intersect in a unique bounded trajectory—the quantum transition state—that anchors such a surface. Quantum mechanics does not destroy the transition state; it becomes quantum.

We classify the distributional self-embeddings of the centered first limit Bourgain-Rosenthal-Schechtman space $R_\omega^{p,0}$, $1<p<\infty$. Using a Boolean rigidity principle for its canonical independent-sum realization, we show that every such embedding is induced by a finite packing of Bernoulli factors. As a consequence, we also prove that $R_\omega^{p,0}$ admits no proper non-zero internal compressions. Moreover, for $p\notin2\mathbb N$, we obtain a complete description of the linear isometric embeddings of the non-centered space $R_\omega^p$, and, for $p\neq2$, we determine its group of surjective linear isometries.

In urban contexts, travel times vary strongly with the time of day and traffic conditions. The Time-Dependent Vehicle Routing Problem with Time Windows (TDVRPTW) extends the classical VRPTW by making travel times depend on departure time, with the objective of minimizing the total travel time of at most $k$ vehicle routes serving customers within strict time windows. Exact resolution of this realistic yet understudied problem currently relies on the Branch & Price approach of Dabia et al. (2013), in which a Column Generation scheme decomposes the problem into a set-partitioning master problem and a resource-constrained shortest-path pricing problem solved by dynamic programming. This work, conducted within the MAMUT project (Machine Learning and Matheuristics for Urban Transport), aims to advance the state of the art for the exact and explainable resolution of the TDVRPTW. We propose to reimplement and revisit the pioneering Branch & Price algorithm by transferring recent advances made on the single-vehicle variant (TDTSPTW) to the multi-vehicle setting, notably an exact anytime extension of A* and a Large Neighborhood Search guided by dynamic programming for the pricing sub-problem. We further explore the integration of data mining and machine learning to guide column generation by exploiting knowledge from previously computed routes. The goal is a high-performance and interpretable solver that meets the requirements of real-world urban transport while preserving guarantees on solution quality.

The Vehicle Routing Problem with Time Windows (VRPTW) requires a fleet of capacitated vehicles to serve customers within strict time windows while minimizing total travel time. Constructive heuristics are fundamental for generating solutions and commonly serve as starting points for metaheuristics such as Iterated Local Search (ILS), Genetic Algorithms, and Ant Colony Optimization (ACO). This work studies three classical constructive heuristics - Nearest Neighbor, Best Insertion, and Regret-k - within a randomized, multi-start framework, a setting that remains largely under-explored despite their well-established deterministic use. We address how to effectively randomize these heuristics and, in particular, propose a novel randomization of Regret-k by separating the ”who” (which customer to insert next) and ”where” (insertion position) decisions, applying a probabilistic selection proportional to the regret value to preserve the heuristic's foresight while injecting the diversity needed for a multi-start approach. We further study the impact of combining these greedy randomized constructions with Local Search and ACO. Experiments follow the DIMACS 2021 conventions (integer, truncated Euclidean distances) on the Solomon and Gehring & Homberger benchmarks (100–1000 customers), comparing against the Best-Known Solutions and the state-of-the-art Hybrid Genetic Search solver, with all critical components implemented in modern C++ and bound to Python via pybind11. To the best of our knowledge, the randomization of Regret-k has never been examined; our analysis highlights the trade-off between solution quality and computational cost, providing a preliminary step toward hybrid methods (ACO, ILS, Branch-and-Price) for the Time-Dependent VRPTW within the MAMUT project.

This paper studies the optimal control problem of a queueing system with three servers: one fast server and two identical slow servers. The two-server version of this problem, with one fast server and one slow server, was introduced by Lin and Kumar (1984), and the optimal policy has been shown to be of threshold type. However, generalizing this result beyond the two-server setting has been considered an open problem. In this paper, we resolve the first nontrivial case by proving that a threshold policy is optimal for the three-server system considered. The core technical ideas in this paper are generated by GPT-5.5 Pro. We have included a short report describing the authors' interactions with GPT-5.5 Pro. The authors verified the proofs and rewrote the paper for better rigor, clarity, and exposition. In addition, three key lemmas have also been verified in Lean 4.

Identification conditions describe the computability of a target query or parameter of interest as a function of the type and amount of information available. In causal identification, this information is often expressed in the form of a causal graph, and data are observed or collected for some subset of variables in the graph. Target queries may be for a single effect alone or for a class of effects in a given model. The derivation of an identification algorithm then defines mathematically the process by which the desired causal effect(s) can be uniquely determined, theoretically, in expectation. Identifiability in expectation, or 'theoretical identifiability,' generally assumes asymptotic properties, infinite data, or other mathematically idealized conditions. In this paper, we explore a fundamental distinction between this theoretical, idealized notion of identifiability and a proposed alternative that is computation-bound. The framework we propose - 'computational identifiability' - is to instead define a finite computational search procedure for an empirical estimator. If this process finds an estimator empirically, within a desired error tolerance, then identifiability is satisfied, conditional on the specified assumptions of the search (i.e., a prior distribution over the parameters) and conditional on the search procedure itself. Through several experiments, we demonstrate how this framework allows us to answer fine-grained, practical identification questions, such as identification with small finite samples, with ambiguous graphical criteria, with mixed observational-interventional data, and across counterfactual data and estimands. Code is available at https://github.com/lbynum/metadentify.

Real-time autonomous navigation in dynamic, unknown environments remains a fundamental challenge for mobile robotics. We propose a map-free framework that tightly integrates reactive rolling-horizon planning with nonlinear Model Predictive Control (MPC). At each control cycle, a LiDAR-based Gaussian occupancy representation is constructed and used to generate collision-free trajectories via A* search, which are then tracked by a CasADi/IPOPT MPC formulation incorporating a smooth sigmoid obstacle barrier. To improve robustness to parameter sensitivity, we adopt an offline Bayesian optimization scheme based on Tree-structured Parzen Estimators (TPE), which identifies near-optimal controller parameters with respect to a composite navigation objective. In addition, a Gaussian Process surrogate is used to analyze parameter sensitivity and provide insight into the optimization landscape. The proposed framework is robot-agnostic and is evaluated on the Unitree Go2 quadruped in simulation using Gazebo, followed by deployment on the physical robot. Experimental results show that parameters tuned in simulation transfer effectively to hardware, maintaining comparable performance without additional tuning. The full system achieves up to a 90.0% navigation success rate when deployed, along with a 38.9% average improvement in the evaluation metrics across simulated environments.

We study the von Neumann algebra generated by a quasi-product state associated with the one-dimensional hard-core lattice gas. If \(\kappa>0\) denotes the activity (fugacity), the transfer-matrix construction gives the stationary transition matrix \[ P=\begin{pmatrix}p&q\\ 1&0\end{pmatrix}, \qquad p+q=1, \qquad \kappa=\frac{q}{p^2}. \] The stationary Markov measure on the hard-core path space induces a faithful diagonal state on the corresponding path \(AF\)-algebra. We give a state-preserving identification of its GNS von Neumann algebra with the Feldman–Moore algebra of the measured tail equivalence relation. The relation is hyperfinite and ergodic, and its nonzero ratio set is \(\kappa^{\mathbb Z}\). Consequently, the generated algebra is the hyperfinite factor of type \(\mathrm{II}_1\) when \(\kappa=1\), and of type \(\mathrm{III}_{\lambda}\), where \(\lambda=\min\{\kappa,\kappa^{-1}\}\), when \(\kappa\ne1\). The point \(\kappa=1\) is not a thermodynamic phase transition: the pressure and particle density are analytic for all \(\kappa>0\). We also describe the precise support-corner relation with the GNS algebra of the diagonal Gibbs state on the quasi-local spin algebra, and we determine the centralizer and the flow of weights.

The Schramm-Loewner Evolution (SLE) describes a family of fractal curves that arise in the study of the scaling limits of many planar Statistical Physics models. These curves are modeled using the Loewner Differential Equation for the conformal maps $g_t(z)$ with a Brownian motion driver. Using Euler's Method, in the current work we performed numerical experiments to study at a fixed time the quantities $|g_t(z) - \overline{g_t(z)}|$ and $Re(g_t(z)) - Re(\overline{g_t(z)})$, where $Re$ denotes the real part and $\overline{g_t(z)}$ refers to the sample average. These random variables measure the 'spread' of the dynamics from the average behavior at fixed time. One of the scopes of this work is to give numerical predictions for future theoretical investigations on these quantities. When investigating these quantities in the SLE case our experiments predict that the distribution is bimodal when the dynamics started close to the origin, and it can become bell-shaped if the dynamics is started further from the origin. In the second part, we performed experiments for a Multiple SLE model whose driver is Dyson Brownian Motion. Due to singularity in the dynamics of the drivers and the many data points needed, this part is challenging from a computational perspective. In the multiple SLE case, our experiments predict that the distribution is bell-shaped in all cases. In addition, we check the changes in the distributions as we vary the parameter $\kappa$ in the SLE case and $\beta$ in the Multiple SLE case.

We consider the enumeration of $N$-ary convex structures on finite sets. Our main result shows that the total number of convexities is expressed as the binomial transform of the sequence of numbers of grounded convexities. We present the exact numbers of all $N$-ary and grounded $N$-ary convexities for $|X|\leqslant 5$. The obtained integer sequences have been cross-referenced with the OEIS. As a result, we identified both known sequences (such as A000798) and new ones not previously listed.

Using a sequence of high-school level mathematics questions that were not available on the Internet, we compare the performance of the popular AI software Claude with that of my friends and fellow human beings Pierre and Catherine. Pierre had solid scientific training as a young man, while Catherine studied literature. All three were subjected to a simulated pre-calculus oral exam with main questions and follow-up questions. Their performances are compared and the ones with the best and worst performances are identified. The outcome is that the current version of Claude, even though it is an extremely useful tool that has probably recorded the solution to nearly all calculus questions that are available on the Internet, exhibits only a very limited understanding of the subject and does not exhibit the ability to make intelligent connections between different features of a pre-calculus mathematics problem that it has never seen before.

“`text Let $K_1=B[0,1]$ be the trace of a planar Brownian motion, and let $N_{\mathrm{in}}(\eta)$ be the number of its cavities, the bounded components of $\mathbb{R}^2\setminus K_1$, whose inradius exceeds $\eta$. We prove that, almost surely and in $L^p$ for every $p\in[1,2]$, $\eta^2(\log(1/\eta))^2 N_{\mathrm{in}}(\eta)\to c_{\mathrm{in}}=\frac{\pi}{2}\int G^2 d\mathcal{L}\in(0,\frac12]$, where $G$ is the ratio of the inradius of a shape to the square root of its area and $\mathcal{L}$ is Werner's limiting shape law. The inradius, and not the area by which the cavities have been counted classically, decides survival under thickening: a cavity contains a hole of the Wiener sausage of radius $r$ if and only if its inradius exceeds $r$. Through Brownian scaling, the number $\beta_1^T(r)$ of holes of the sausage of radius $r$ of $B[0,T]$, the Betti curve of its degree-one persistent homology, is bounded below at the scale $T/(r^2(\log T)^2)$, with the identified constant $4c_{\mathrm{in}}$ in expectation and with any smaller constant with probability tending to one. A matching upper bound cannot follow from any general bound in terms of perimeter, area, diameter or length; we prove that it is equivalent to a first-moment estimate for near-critical pockets, the holes at level $\eta$ on which the distance to the trace is at most $2\eta$. That estimate is proved in a companion paper, so that the expected number of holes is of exact order $T/(r^2(\log T)^2)$. “`

The concept of downward conditional monotonicity for the Markov-modulated Poisson process (MMPP) is introduced and used to derive the optimal stochastic domination of a standard Poisson point process. The maximum arrival rate for the Poisson process which allows this domination to exist is shown to be related to an eigenvalue extracted from the generator matrix of the quasi-birth–death (QBD) formulation of the MMPP. This allows derivation of survival and extinction regimes for a large family of contact processes whose infection and recovery rates vary over time according to an underlying random environment with a finite number of states. Direct comparison with standard contact processes which dominate from above and below accomplishes this.

In this paper, we consider a Gorenstein-dimensional analogue of Foxby's characterization of Gorenstein rings. We prove that a commutative Noetherian local ring is Gorenstein if it admits a complex whose depth, Gorenstein flat dimension, and Gorenstein injective dimension are all finite. This gives an affirmative answer to the original question of Christensen, Foxby, and Holm, which had remained open in this generality even for modules, and at the same time establishes its natural extension to complexes.

We consider an inverse boundary value problem for the time-independent Schrödinger equation in dimension three. We prove that the local Dirichlet-to-Neumann map defined near a boundary point uniquely determines the potential in a neighborhood of the boundary point in the interior. In particular, we show that the uniqueness question can be reduced to the injectivity of a weighted X-ray transform, which links inverse boundary value problems to integral geometry.

Conditional on the Generalized Riemann Hypothesis for $L(s,\chi)$, we prove the Selberg–Tsang high-moment bound for $X_\chi(t) = \operatorname{Im}\log L(\tfrac12+it,\chi)$ at fixed squarefree odd conductor $q \ge 3$ and primitive non-principal character $\chi$. Writing $L_T = \log\log(qT)$: for every $K > 0$ there exist constants $C_K$ and $T_0$ such that $\frac{1}{T}\int_T^{2T} |X_\chi(t)|^{2k}\,dt \le (C_K\,k\,L_T)^k$ for all $T \ge T_0$ and every integer $1 \le k \le K L_T$. The proof ports Selberg's pointwise approximate formula for $S(t)$ to $L(s,\chi)$ at fixed conductor under GRH, splits it into three prime-power Dirichlet polynomials, and evaluates their moments via Soundararajan's mean-value lemma. As a corollary, Markov's inequality yields a Gaussian-scale tail $\exp(-c V^2 / L_T)$ for $\sqrt{L_T} \ll V \ll L_T$ – a GRH-conditional, fixed-conductor, imaginary-part analogue of the large-deviation upper bounds known for $\log|\zeta(\tfrac12+it)|$.

We classify, up to almost diffeomorphism, the smooth closed oriented manifolds homotopy equivalent to each of the sphere products $S^{4k-1}\times S^{4k}$, $S^{4k}\times S^{4k}$, and $S^{4k}\times S^{4k+1}$. In each case we realize the image of the normal-invariant map in the smooth surgery exact sequence by explicit families of manifolds: sphere bundles over $S^{4k}$; pinch maps and Milnor plumbings of disk bundles; and Novikov sphere bundles together with connected sums of homotopy spheres.

We consider a two-dimensional piecewise $C^2$ domain that cuts through a quasi-uniform fixed polygonal background mesh, for instance made of quadrilaterals. A simple procedure based on convex hulls gives rise to a rather small number of polygonal boundary elements of various shapes, including elements with small edges and large aspect ratios; this is the computational mesh for a virtual element method (VEM), a trimmed background mesh. We classify all possible geometric configurations and study their stability and approximability properties. This entails deriving robust stabilization mechanisms and interpolation estimates for anisotropic elements and elements with small cuts, as well as a weak maximum principle for enhanced virtual elements; these contributions have intrinsic interest for VEM theory on geometric flexibility. We prove that the resulting VEM is uniformly stable in $H^1$, and also show optimal order-regularity error estimates in $H^1$ and $L^2$. Insightful numerical experiments corroborate and complement our theory. The proposed method is suitable for treating ALE formulations of problems in moving domains.